Alternating series; first term is 0. Do I have a problem?

You can remove a finite number of terms and not affect convergence.

Observe that your series just rewrites $$ \sum_{n=1}^\infty (-1)^n \frac{n^2-1}{n^3+1}=\sum_{n=\color{red}{2}}^\infty (-1)^n \frac{n^2-1}{n^3+1}. $$