Testing A Series For Convergence

Determine whether the series
$\sum_{n=0}^{\infty} \frac{3n^2 + 2n + 1}{n^3 + 1}$ with n from 0 to infinity
converges or diverges.

So far I thought about dividing the numerator by the denominator, but that got very messy.

I thought about comparing that to the series of $\frac{1}{k^3 + 1} but then I got stuck.

Also, a related question.
A theorem states that if the limit of a series as n approaches infinity is not equal to zero, the series diverges. However it states that the series is from n=1 to infinity. Would it also apply in this case where it goes from n=0 to infinity?

Thanks!