how would I test this series for convergence of divergence?
the series from n=0 to infinity of
(n^2+1)/(n^3+1)
You can do the Integral test. You see that the Series diverges logarithmically.
Exercise: Try to compute the large N asymptotics of
n=0 to N of
(n^2+1)/(n^3+1)
For large N the summation will be:
1/3 Log(N) + a + b/N + c/N^2 + ...
Try to find out what a, b, c etc. are.