Summing series

I am not sure how to find

\sum \limits _{r=1}^n \frac{r^2}{2^r}

. There is an explicit formula for this partial sum on wolfram, but I want to know to find it.

I am not sure how to find

\sum \limits _{r=1}^n \frac{r^2}{2^r}

. There is an explicit formula for this partial sum on wolfram, but I want to know to find it.

Let the nth partial sum be

S_n

. Express

S_n' = \displaystyle\sum_{r=1}^n \dfrac{d}{dr} \left[ \dfrac{r^2}{2^r} \right]

in terms of

S_n

and another sum

T_n

. Then, by differentiating again and using what you know about

T_n

in terms of

S_n

and

S_n'

, obtain a 2nd order linear ODE in

S_n

to solve.

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