Sr Examen
Otras calculadoras
- Serie de Taylor paso a paso
- Serie de Fourier paso a paso
- Ecuaciones diferenciales paso a paso
- Producto de la serie
-
¿Cómo usar?
- Suma de la serie:
-
(n+1)/n
-
(n+1)/3^n
-
6/(9n^2+12n-5)
-
(7/8)^n
-
Expresiones idénticas
- x^n/(uno +x^2n)
- x en el grado n dividir por (1 más x al cuadrado n)
- x en el grado n dividir por (uno más x al cuadrado n)
- xn/(1+x2n)
- xn/1+x2n
- x^n/(1+x²n)
- x en el grado n/(1+x en el grado 2n)
- x^n/1+x^2n
- x^n dividir por (1+x^2n)
-
Expresiones semejantes
- x^n/(1-x^2n)
- (x^n)/1+x^(2*n)
- (x^n)/(1+x^(2*n))
Suma de la serie x^n/(1+x^2n)
Solución
Ha introducido
[src]
oo ____ \ ` \ n \ x ) -------- / 2 / 1 + x *n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{x^{n}}{n x^{2} + 1}$$
Sum(x^n/(1 + x^2*n), (n, 1, oo))
Radio de convergencia de la serie de potencias
Se da una serie:
$$\frac{x^{n}}{n x^{2} + 1}$$
Es la serie del tipo
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- serie de potencias.
El radio de convergencia de la serie de potencias puede calcularse por la fórmula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
En nuestro caso
$$a_{n} = \frac{1}{n x^{2} + 1}$$
y
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
entonces
$$R = \lim_{n \to \infty} \left|{\frac{x^{2} \left(n + 1\right) + 1}{n x^{2} + 1}}\right|$$
Tomamos como el límite
hallamos
$$R^{1} = 1$$
$$R = 1$$
$$\frac{x^{n}}{n x^{2} + 1}$$
Es la serie del tipo
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- serie de potencias.
El radio de convergencia de la serie de potencias puede calcularse por la fórmula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
En nuestro caso
$$a_{n} = \frac{1}{n x^{2} + 1}$$
y
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
entonces
$$R = \lim_{n \to \infty} \left|{\frac{x^{2} \left(n + 1\right) + 1}{n x^{2} + 1}}\right|$$
Tomamos como el límite
hallamos
$$R^{1} = 1$$
$$R = 1$$
Respuesta
[src]
/ / 2\ | | 1 + x | |lerchphi|x, 1, ------| | | 2 | / / / 2 2 \ / 2 2 \ 2 2 \ / / 2 2 \ / 2 2 \ 2 2 \ / / 2 2 \ / 2 2 \ 2 2 / 2 2 \ / 2 2 \ 2 2 \\ | \ x / | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | | | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) | im (x) re (x) | / 2 2 \ | re (x) im (x) | / 2 2 \ 4*im (x)*re (x) || |---------------------- for Or|And||x| <= 1, 1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ < 0|, And|1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ >= 1, |x| < 1|, And|1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ >= 0, |x| <= 1, 1 + |------------------ - ------------------|*\1 - 2*im (x) + 2*re (x)/ + |------------------ - ------------------|*\1 + re (x) - im (x)/ - ------------------ < 1, x != 1|| | x | | | 2 2| | 2 2| 2 | | | 2 2| | 2 2| 2 | | | 2 2| | 2 2| 2 | 2 2| | 2 2| 2 || | | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ | | |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ |/ 2 2 \ / 2 2 \ | |/ 2 2 \ / 2 2 \ | / 2 2 \ || | \ \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ / \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ / \ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ \\im (x) + re (x)/ \im (x) + re (x)/ / \\im (x) + re (x)/ \im (x) + re (x)/ / \im (x) + re (x)/ // | < oo | ____ | \ ` | \ n | \ x | ) -------- otherwise | / 2 | / 1 + n*x | /___, | n = 1 \
$$\begin{cases} \frac{\Phi\left(x, 1, \frac{x^{2} + 1}{x^{2}}\right)}{x} & \text{for}\: \left(\left|{x}\right| \leq 1 \wedge \left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} < 0\right) \vee \left(\left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \geq 1 \wedge \left|{x}\right| < 1\right) \vee \left(\left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \geq 0 \wedge \left|{x}\right| \leq 1 \wedge \left(- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(2 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + \left(\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right) \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right) + 1 - \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} < 1 \wedge x \neq 1\right) \\\sum_{n=1}^{\infty} \frac{x^{n}}{n x^{2} + 1} & \text{otherwise} \end{cases}$$
Piecewise((lerchphi(x, 1, (1 + x^2)/x^2)/x, ((|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 0))∨((|x| < 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 1))∨((Ne(x, 1))∧(|x| <= 1)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 >= 0)∧(1 + (im(x)^2/(im(x)^2 + re(x)^2)^2 - re(x)^2/(im(x)^2 + re(x)^2)^2)*(1 - 2*im(x)^2 + 2*re(x)^2) + (re(x)^2/(im(x)^2 + re(x)^2)^2 - im(x)^2/(im(x)^2 + re(x)^2)^2)*(1 + re(x)^2 - im(x)^2) - 4*im(x)^2*re(x)^2/(im(x)^2 + re(x)^2)^2 < 1))), (Sum(x^n/(1 + n*x^2), (n, 1, oo)), True))
Ejemplos de hallazgo de la suma de la serie
- Suma de la serie de potencias
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Serie de Flint Hills
csc(n)^2/n^3
- Serie de cuadrados inversos
1/n^2
1/n^4
1/n^6
- Serie armónica
1/n
- Serie de Grandi
(-1)^n
- Serie alternada
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Serie de Newton-Mercator
(-1)^(n + 1)/n*x^n
- Investigar la convergencia de la serie
(3*n - 1)/(-5)^n