TITLE: Sum of Infinite series with a Geometric series in multiply
QUESTION [8 upvotes]: I came across these series while solving a probability question.
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let |r| < 1 ,
$$S_n=\sum_{k=0}^{\infty}k^n.r^k$$
For n=0 ,it's a GP.    $S_0=\frac{1}{1-r}$
For n=1 ,it's a AGP , $S_1=\frac{-1}{1-r}+\frac{1}{(1-r)^2}$
for n=2 , This series can be reduced to AGP by substituting $k^2=1.3.5....(2k-1)$  & $S_2=\frac{-r}{(1-r)^2}+\frac{2r}{(1-r)^3}$.
Is it possible to find sum further in this series . Is there any pattern.

REPLY [5 votes]: Given $S_n=\sum_{k=0}^{\infty}k^n\cdot r^k$, it must be:
$$\begin{align}S_0&=\sum_{k=0}^{\infty}r^k=\frac1{1-r}; \\ S_0'&=\frac{1}{(1-r)^2}=\color{red}1S_0^2;\\
S_1&=\sum_{k=0}^{\infty}k\cdot r^k=rS_0'=rS_0^2; \\  
S_1'&=\color{red}1S_0^2+\color{red}2rS_0^3\\
S_2&=\sum_{k=0}^{\infty}k^2\cdot r^k=rS_1'=rS_0^2+2r^2S_0^3; \\ S_2'&=\color{red}1S_0^2+\color{red}6rS_0^3+\color{red}6r^2S_0^4\\
S_3&=\sum_{k=0}^{\infty}k^3\cdot r^k=rS_2'; \\ S_3'&=\color{red}1S_0^2+\color{red}{14}rS_0^3+\color{red}{36}r^2S_0^4+\color{red}{24}r^3S_0^5\end{align}$$
where the coefficients in red are the number sequence A019538.