Question

how would I test this series for convergence of divergence?

the series from n=0 to infinity of

(n^2+1)/(n^3+1)

Answer 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.