# Convergence and divergence of sum

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.

• NSum[(8 n + Sqrt[n])/(n^4 - n^2 + 5), {n, 1, Infinity}] -> 3.50589 – user36273 Nov 26 '16 at 11:18
• Related: (126839). – corey979 Nov 26 '16 at 16:31

This should do it:

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


It returns True.

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.

• Your answer is not related to Mathematica. – Mirko Aveta Nov 26 '16 at 19:41