Is this alternating series convergent?
If your professor gave you this problem then it may be a typo. Perhaps he means $$ \sum_{n=2}^\infty\frac{(-1)^n}{n-1} $$ which is the same as $$ \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}. $$ This series is conditionally convergent by the AST and $p$-tests, but with the typo intact it is neither convergent nor divergent, but rather undefined.