# Sum convergence test problem.

I’ve come across this sum, and I need to test the convergence, but I really have no idea how to get started. This is the sum: $$\sum_{n=1}^{\infty} \frac{n^{\sqrt{6}+1}+n+5}{n^{\sqrt6+2}+2n+10}$$ Can you give me some tips? Thanks!.

-

Hint: $\left(\dfrac{n^{\sqrt{6}+1}+n+5}{n^{\sqrt6+2}+2n+10}\right)_{n\in \mathbb N}\sim _\infty\left(\dfrac{n^{\sqrt{6}+1}}{n^{\sqrt6+2}}\right)_{n\in \mathbb N}=\,?$

-
I get it, with comparision test my sum does not converge. Thank you. – user137209 Apr 27 '14 at 9:57
@user137209 No need to hurry in accepting my answer. Maybe you like another one better. – Git Gud Apr 27 '14 at 9:59

For $n\ge1$, \begin{align} \frac{n^{\sqrt{6}+1}+n+5}{n^{\sqrt6+2}+2n+10} &=\frac1n\frac{1+n^{-\sqrt6}+5n^{-\sqrt6-1}}{1+2n^{-\sqrt6-1}+10n^{-\sqrt6-2}}\\ &\ge\frac1n\cdot\frac7{13} \end{align}

-