Testing a Series for Convergence or Divergence

Question

The problem is to test the following series for convergence:

$\displaystyle\sum\limits_{i=1}^{\infty} ((i^2+1)/(i^3+1))$

I tried several tests like the root and limit comparison test but they yielded no usable results. I found online that this series would diverge by the comparison test, however, I am not sure what series I would compare this to. Can anyone point me in the right direction?

Answer 1

The series is divergent as one can compare with the divergent series $\sum \dfrac{1}{i} $:

$$ \dfrac{1+i^2 }{1+i^3} > \frac{i^2}{i^3 + i^3} = \frac{1}{2i} $$

Answer 2

You may use the limit comparison test with the sequence $a_i = \frac{1}{i}$.

https://en.m.wikipedia.org/wiki/Limit_comparison_test

Answer 3

The simplest way uses an asymptotic equivalent:

We know that a polynomial is asymptotically equivalent to its leading, and the equivalence of functions is compatible with multiplication and division, so $i^2+1\sim_\infty i^2$, $i^3+1\sim_\infty i^3$, whence $$\frac{i^2+1}{i^3+1}\sim_\infty\frac{i^2}{i^3}=\frac1 i,$$ and the latter series diverges, therefore the given series diverges too.