Sr Examen
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- Límite de la función:
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Límite de ((-1+2*x)/(1+2*x))^(3*x)
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Límite de (-1+2^(cos(x)^2))/log(sin(x))
-
Límite de (2^n+3^n)/(2^n-3^n)
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Límite de (1-x)/(-1+x^2)
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Expresiones idénticas
- log(n)/log(factorial(tres *n)*factorial(- uno + tres *n))
- logaritmo de (n) dividir por logaritmo de (factorial(3 multiplicar por n) multiplicar por factorial( menos 1 más 3 multiplicar por n))
- logaritmo de (n) dividir por logaritmo de (factorial(tres multiplicar por n) multiplicar por factorial( menos uno más tres multiplicar por n))
- log(n)/log(factorial(3n)factorial(-1+3n))
- logn/logfactorial3nfactorial-1+3n
- log(n) dividir por log(factorial(3*n)*factorial(-1+3*n))
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Expresiones semejantes
- log(n)/log(factorial(3*n)*factorial(-1-3*n))
- log(n)/log(factorial(3*n)*factorial(1+3*n))
-
Expresiones con funciones
- Logaritmo log
- log(k)
- log((5+x)/x)^(5+3*x)/log(10)
- log(log(x))/(-e+5*x)
- log(x)/(1-3*log(sin(x)))
- log(sin(6*x))/log(sin(4*x))
- Logaritmo log
- log(k)
- log((5+x)/x)^(5+3*x)/log(10)
- log(log(x))/(-e+5*x)
- log(x)/(1-3*log(sin(x)))
- log(sin(6*x))/log(sin(4*x))
Límite de la función log(n)/log(factorial(3*n)*factorial(-1+3*n))
Solución
Ha introducido
[src]
/ log(n) \ lim |-----------------------| n->oo\log((3*n)!*(-1 + 3*n)!)/
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
Limit(log(n)/log(factorial(3*n)*factorial(-1 + 3*n)), n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty} \log{\left(n \right)} = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} \log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} \log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(3 n\right)! \left(3 n - 1\right)!}{n \left(3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} + 3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n}}{\frac{d}{d n} \left(3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} + 3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n} + \frac{3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n^{2}}}{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} + 9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n} + \frac{3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n^{2}}\right)}{\frac{d}{d n} \left(9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} + 9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n} + \frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{6 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{2}} - \frac{6 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{2 \left(3 n\right)! \left(3 n - 1\right)!}{n^{3}}}{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n} + \frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{6 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{2}} - \frac{6 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{2 \left(3 n\right)! \left(3 n - 1\right)!}{n^{3}}\right)}{\frac{d}{d n} \left(27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{2}} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{54 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{18 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{3}} + \frac{18 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{6 \left(3 n\right)! \left(3 n - 1\right)!}{n^{4}}}{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} + 486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{2}} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{54 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{18 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{3}} + \frac{18 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{6 \left(3 n\right)! \left(3 n - 1\right)!}{n^{4}}\right)}{\frac{d}{d n} \left(81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} + 486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)}}{n} + \frac{486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n^{2}} - \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{3}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{3}} + \frac{216 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{72 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{4}} - \frac{72 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{4}} + \frac{24 \left(3 n\right)! \left(3 n - 1\right)!}{n^{5}}}{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{5}{\left(0,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 1215 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(4,3 n \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{5}{\left(0,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 3645 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(4,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)}}{n} + \frac{486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n^{2}} - \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{3}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{3}} + \frac{216 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{72 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{4}} - \frac{72 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{4}} + \frac{24 \left(3 n\right)! \left(3 n - 1\right)!}{n^{5}}}{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{5}{\left(0,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 1215 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(4,3 n \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{5}{\left(0,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 3645 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(4,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
oo/oo,
tal que el límite para el numerador es
$$\lim_{n \to \infty} \log{\left(n \right)} = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} \log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(n \right)}}{\frac{d}{d n} \log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(3 n\right)! \left(3 n - 1\right)!}{n \left(3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} + 3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n}}{\frac{d}{d n} \left(3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} + 3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n} + \frac{3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n^{2}}}{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} + 9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{3 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n} + \frac{3 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{\left(3 n\right)! \left(3 n - 1\right)!}{n^{2}}\right)}{\frac{d}{d n} \left(9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} + 9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n} + \frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{6 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{2}} - \frac{6 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{2 \left(3 n\right)! \left(3 n - 1\right)!}{n^{3}}}{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n} + \frac{9 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{9 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{18 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} - \frac{6 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{2}} - \frac{6 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{2 \left(3 n\right)! \left(3 n - 1\right)!}{n^{3}}\right)}{\frac{d}{d n} \left(27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{2}} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{54 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{18 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{3}} + \frac{18 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{6 \left(3 n\right)! \left(3 n - 1\right)!}{n^{4}}}{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} + 486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{2}} - \frac{27 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{27 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{54 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} + \frac{18 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{3}} + \frac{18 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{6 \left(3 n\right)! \left(3 n - 1\right)!}{n^{4}}\right)}{\frac{d}{d n} \left(81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} + 486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)}}{n} + \frac{486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n^{2}} - \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{3}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{3}} + \frac{216 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{72 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{4}} - \frac{72 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{4}} + \frac{24 \left(3 n\right)! \left(3 n - 1\right)!}{n^{5}}}{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{5}{\left(0,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 1215 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(4,3 n \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{5}{\left(0,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 3645 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(4,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)}}{n} + \frac{486 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(1,3 n \right)}}{n} + \frac{81 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(3,3 n \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)}}{n} + \frac{81 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(3,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{972 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n} + \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}{n} + \frac{486 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)}}{n^{2}} - \frac{324 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} - \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(2,3 n \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(2,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{2}} - \frac{324 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)}}{n^{2}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)}}{n^{3}} + \frac{108 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)}}{n^{3}} + \frac{108 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)}}{n^{3}} + \frac{216 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{3}} - \frac{72 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)}}{n^{4}} - \frac{72 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)}}{n^{4}} + \frac{24 \left(3 n\right)! \left(3 n - 1\right)!}{n^{5}}}{243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{5}{\left(0,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 1215 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 243 \left(3 n\right)! \Gamma\left(3 n\right) \operatorname{polygamma}{\left(4,3 n \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{5}{\left(0,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 3645 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 243 \left(3 n - 1\right)! \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(4,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{4}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{4}{\left(0,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 4860 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}^{2}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n \right)} \operatorname{polygamma}{\left(3,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{3}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}^{2}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)} + 3645 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,3 n \right)} + 7290 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(1,3 n + 1 \right)} + 1215 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(0,3 n + 1 \right)} \operatorname{polygamma}{\left(3,3 n \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n \right)} \operatorname{polygamma}{\left(2,3 n + 1 \right)} + 2430 \Gamma\left(3 n\right) \Gamma\left(3 n + 1\right) \operatorname{polygamma}{\left(1,3 n + 1 \right)} \operatorname{polygamma}{\left(2,3 n \right)}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = 0$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = 0$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = \frac{\infty}{\log{\left(\left(-\infty\right)!^{2} \right)}}$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right)$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = 0$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = 0$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{\log{\left(n \right)}}{\log{\left(\left(3 n\right)! \left(3 n - 1\right)! \right)}}\right) = \frac{\infty}{\log{\left(\left(-\infty\right)!^{2} \right)}}$$
Más detalles con n→-oo