Given the following series Wolfram Alpha says it diverges but when I was trying to prove it I came to the conclusion it converges and I can't figure out what's wrong. $$\sum_{n=2}^\infty \frac{(-1)^n}{n+(-1)^n}$$
$$S_{2n+1}=\frac{1}{3}-\frac{1}{2}+\frac{1}{5}-\frac{1}{4}+ ... +\frac{1}{2n+1}-\frac{1}{2n}$$
$$S_{2n+1}=-\frac{1}{6}-\frac{1}{20}-...-\frac{1}{2n(2n+1)}=-\sum\limits_{k=2}^{n} \frac{1}{2n(2n+1)} \to L$$
Where L is the sum of this clearly convergent series.
$$S_{2n+2}=S_{2n+1}+\frac{1}{2n+3} \to L$$ $$S_{n}\to L$$
So surely this must converge? I saw something similar used when proving the alternating series test but couldn't apply it on this series since $\frac{(-1)^n}{n+(-1)^n}$ isn't monotonous.
Edit: Seems like I must have messed up my input as I can't replicate the error on Wolfram. Thanks for the answers and sorry for wasting your time. Needed this result for something so got freaked out when it said it diverges the first time.