# Numerical sum problem.

I just started the series chapter and I come across some series that I don't know how should I resolve them. All of them have the same structure : $$\sum_{n=0}^{\infty} (-1)^n ...$$ For example: $$\sum_{n=0}^{\infty} (-1)^n \frac{n+1}{2^n} or$$ $$\sum_{n=0}^{\infty} (-1)^n \frac{n^2}{2^n}$$ Some tips on how should i aproache this would be greatly appreciated.

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I edited my post. Sorry it's n from 0 to infinity. – user137209 Apr 26 '14 at 13:44
I suggest you change the upper value to $m$ – Claude Leibovici Apr 26 '14 at 13:45

First, absorb $(-1)^n$ with $\displaystyle\frac1{2^n}$ to find

$$\sum_{n=0}^{\infty}(-1)^n\frac{n+1}{2^n}=\sum_{n=0}^{\infty}(n+1)\left(-\frac12\right)^n$$ $$=\sum_{n=0}^{\infty}(n-1)\left(-\frac12\right)^n+2\underbrace{\sum_{n=0}^{\infty}\left(-\frac12\right)^n}_{\text{ Geomtric Series}}$$

For $|r|<1,$ $$\sum_{n=0}^{\infty}(n-1)r^n=\sum_{n=0}^{\infty}(n-1)r^n=\frac{d \sum_{n=0}^{\infty}r^n}{dr}$$

It is funny ! Today, you wrote at least twice $Geomtric$ $Series$. Cheers. – Claude Leibovici Apr 26 '14 at 13:48