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Suma de la serie/ nx^n/3^n(n+1)

Suma de la serie nx^n/3^n(n+1)


∑
=

Solución

Ha introducido [src] 
  oo               
____               
\   `              
 \        n        
  \    n*x         
   )   ----*(n + 1)
  /      n         
 /      3          
/___,              
n = -1
∑n=−1∞nxn3n(n+1)\sum_{n=-1}^{\infty} \frac{n x^{n}}{3^{n}} \left(n + 1\right) 
Sum(((n*x^n)/3^n)*(n + 1), (n, -1, oo))
Respuesta [src] 
//      -3             |x|    \   //       3              |x|    \
||      ---        for --- < 1|   ||       -          for --- < 1|
||       x              3     |   ||       x               3     |
||                            |   ||                             |
||  oo                        |   ||  oo                         |
|< ___                        | + |< ___                         |
|| \  `                       |   || \  `                        |
||  \       -n  n             |   ||  \     -n  2  n             |
||  /    n*3  *x    otherwise |   ||  /    3  *n *x    otherwise |
|| /__,                       |   || /__,                        |
\\n = -1                      /   \\n = -1                       /
{−3xfor ∣x∣3<1∑n=−1∞3−nnxnotherwise+{3xfor ∣x∣3<1∑n=−1∞3−nn2xnotherwise\begin{cases} - \frac{3}{x} & \text{for}\: \frac{\left|{x}\right|}{3} < 1 \\\sum_{n=-1}^{\infty} 3^{- n} n x^{n} & \text{otherwise} \end{cases} + \begin{cases} \frac{3}{x} & \text{for}\: \frac{\left|{x}\right|}{3} < 1 \\\sum_{n=-1}^{\infty} 3^{- n} n^{2} x^{n} & \text{otherwise} \end{cases} 
Piecewise((-3/x, |x|/3 < 1), (Sum(n*3^(-n)*x^n, (n, -1, oo)), True)) + Piecewise((3/x, |x|/3 < 1), (Sum(3^(-n)*n^2*x^n, (n, -1, oo)), True))

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