<s>[INST] find sum of $\sum_{n=1}^{\infty}\frac{(-1)^n}{n-(-1)^n}$ I want to find the exact sum of this expression: $\sum_{n=1}^{\infty}\frac{(-1)^n}{n-(-1)^n}$
I've already proved that it converges by condition. 
Also, I think that it's sort of telescoping series,
because if I open it I get: 
$\frac{-1}{2}+\frac{1}{1}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}+\frac{1}{5}...$
But I cant think of way to calculate it.
 [/INST] Hint. Note that if $N$ is even then
$$\begin{align}S_N&=\sum_{n=1}^{N}\frac{(-1)^n}{n-(-1)^n}=
-\frac{1}{2}+\frac{1}{1}-\frac{1}{4}+\frac{1}{3}+\dots-\frac{1}{N}+\frac{1}{N-1}\\&=
\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots+\frac{1}{N-1}-\frac{1}{N}=\sum_{n=1}^{N}\frac{(-1)^{n+1}}{n},
\end{align}$$
and
$$S_{N+1}=S_N+\frac{(-1)^{N+1}}{N+1-(-1)^{N+1}}=\sum_{n=1}^{N}\frac{(-1)^{n+1}}{n}
-\frac{1}{N+2}.$$
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