# 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!.

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