- #1

Felipe Lincoln

Gold Member

- 99

- 11

## Homework Statement

By finding a closed formula for the nth partial sum

*##s_n##,*

show that the series ## s=\sum\limits_{n=1}^{\infty}nx^n## converges to ##\dfrac{x}{(1-x)^2}## when ##|x|<1## and diverges otherwise.

## Homework Equations

Maybe ##s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}## when ##|x|<1##

## The Attempt at a Solution

Finding the ##s_n##

##s_n + 1=1 + \sum\limits_{k=1}^{n}(\sqrt[k]{k}x)^k = 1 + x + 2x^2 + 3x^3 + \dots +nx^n= \dfrac{1-(\sqrt[n]{n}x)^{n+1}}{1-\sqrt[n]{n}x}##

but I don't know if I can get anywhere from here, tried several ways and had no success.