Series Convergence: \sum_{n=2}^{\infty}\frac{1+n+n^2}{\sqrt{1+n^2+n^6}}

[azatkgz]

Determine whether the series $$\sum_{n=2}^{\infty}a_n$$ converges absolutely,converges conditionally or diverges.If

$$a_n=\frac{1+n+n^2}{\sqrt{1+n^2+n^6}}$$


For several n I get $$a_n>\frac{1}{n}$$ so I decided that this series is divergent.Right?

 

[Dick]

That would do it. But did you prove that inequality? Just looking at 'several n' basically means you are guessing.

 

[Gib Z]

In fact the result can be much weaker, we don't need to prove the inequality for all n, or even that a_n is more than 1/n, just equal. Divide the terms through by n^2 and see what the n-th term as n --> infinity is.