Determine if $\sum_{n=2}^{\infty} \frac{(-1)^n}{n+(-1)^n}$ converges or diverges I'm having a lot of trouble figuring this one out.

Determine if  $\sum_{i=2}^{\infty} \frac{(-1)^n}{n+(-1)^n}$ converges or diverges

Both ratio and root test are inconclusive and I'm at a loss. Can anyone help me?
Let

$$S_n=\sum_{k=2}^n \frac{(-1)^n}{n+(-1)^n}$$

Take the sum of two consecutive terms:

$$\frac{(-1)^{2n}}{2n+(-1)^{2n}}+\frac{(-1)^{2n+1}}{2n+1+(-1)^{2n+1}}=\frac{1}{2n+1}-\frac{1}{2n}=\frac{-1}{2n(2n+1)}$$

Thus $S_{2n+1}$ converges. Since the terms tend to $0$, $S_{2n}$ is also converging, to the same limit.
Terms are alternatively positive and negative, decrease in absolute value and converge to $0$, so the sseries converges (this is the alternating series test), but it does not converge absolutely.