I was willing to determine the sum of following $$S:=1+\frac12-\frac13-\frac14+\frac15+\frac16-\frac17-\frac18+\cdots$$
I tried the following \begin{align*} S=&\sum\limits_{n=0}^\infty (-1)^n\left(\frac{1}{2n-1}+\frac{1}{2n}\right)\\ =&\sum\limits_{n=0}^\infty (-1)^n\left(\int_0^1 x^{2n-2}dx+\int_0^1 x^{2n-1} dx\right)\\ =&\int_0^1 \sum\limits_{n=0}^\infty (-1)^n\left(x^{2n-2}+x^{2n-1}\right)\\ =&\int_0^1 [x^{-2} \sum\limits_{n=0}^\infty (-1)^nx^{2n}+x^{-1} \sum\limits_{n=0}^\infty (-1)^nx^{2n}] \end{align*} and don't know after his what to do . Can you please help me on this regard?
Thanking you in advance