Consider the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$. How to find its value using complex analysis?
I am considering $f(z)=\sum_{n=1}^{\infty} \frac{z^n}{n^2}$. It is absolutely convergent in $|z|<1$. So it is holomorphic and can be differentiated termwisely, namely $f'(z) = \sum_{n=1}^{\infty} \frac{z^{n-1}}{n}$, thus $zf'(z)=\sum_{n=1}^{\infty} \frac{z^n}{n} = \log (1-z)$ where $\log$ takes principal branch such that $\log1=0$. Thus we have $$ f'(z) = \frac{\log (1-z)}{z} $$ I cannot proceed now. Is it possible to use complex analysis theories to tackle this?
Thank you for any help!