Convergence and divergence of sum

Question

How can I find out if this sum converges or diverges?

I believe that is a question quite simple.

Until I looked in the documentation, but I think that I did not know how to do this.

Sum[(8 n + Sqrt[n])/(n^4 - n^2 + 5), {n, 1, Infinity}]

I believe that is not only insert the sum. But i have to use another feature that i do not know what it was.

Answer 1

This should do it:

SumConvergence[(8 n + Sqrt[n])/(n^4 - n^2 + 5), n]

It returns True.

Answer 2

When n is very large, it is like 1/n^3, it is convergence.

Series[(8 n + Sqrt[n])/(n^4 - n^2 + 5), {n, \[Infinity], 2}]

with reutern $O\left(\left(\frac{1}{n}\right)^3\right)$, so it is convergence.