calculus - Evaluate $sum ln left(1+frac{(-1)^n}{n}right)$

I tried to calculate the sum of the following series without success, any clue would be helpful!
$$\sum_{n=2}^\infty \ln\left(1+\frac{(-1)^n}{n}\right) $$

Answer

Hint:
$$\prod_{n=2}^{2N}\left(1+\frac{(-1)^n}{n}\right)=\prod_{m=1}^{N}\left(1+\frac{1}{2m}\right)\prod_{m=1}^{N-1}\left(1-\frac{1}{2m+1}\right)=\frac{2N+1}{2N}=1+\frac{1}{2N}. $$