How do I find the sum of this series? $$ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} $$ The approach I wanted to use is to find a power series that can become this number series for a certain value of $x$. The problem I encountered is that I am not sure how to extract that $x$. Here is what I tried but I am not sure if this is even valid:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n-1}} = 2\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)2^{n}} => 2\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)} \frac{1}{2^n} => 2\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n}{(n+1)}x^n $$ Where $x=\frac{1}{2}$
Is what I have done so far correct and if it is what should I do next?
