<s>[INST] Help with summation: $\sum_{k=1}^\infty\frac{k(k+2)}{15^k}$ How can one evaluate the below sum? Any help would be greatly appreciated.
$$\sum_{k=1}^\infty\frac{k(k+2)}{15^k}$$
 [/INST] We can built the formula, bottom up.
Let $s_n = \sum_{k=1}^\infty k^n r^k $ 
$n-(n-1)=1 \Rightarrow (1-r)s_1 = s_0$
$n^2-(n-1)^2=2n-1 \Rightarrow (1-r)s_2 = 2 s_1 - s_0$
$s_0 = r + r^2 + r^3 + \cdots =  \frac{r}{1-r}$
$s_1 = \frac{s_0}{1-r} = \frac{r}{(1-r)^2}$
$s_2 = \frac{2 s_1 - s_0}{1-r} = \frac{r(r+1)}{(1-r)^3}$ 
For $r=\frac{1}{15}, s_0=\frac{1}{14}, s_1=\frac{15}{196}, s_2=\frac{30}{343}$
$$ \sum_{k=1}^\infty \frac{k(k+2)}{15^k} = s_2 + 2 s_1 = \frac{165}{686} $$
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