Numerical sum problem.

Question

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.

Answer 1

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

See also: Elementary problems and this

Can you manage the last problem from here?