real analysis - Does the series $sum^{infty}_{n=1}frac{2n^{5}+13n^{3}}{n^{frac{1}{n}}(n^6-n^2+7)}$ converge or diverge? Justify.

Does the series $$\sum^{\infty}_{n=1}\frac{2n^{5}+13n^{3}}{n^{\frac{1}{n}}(n^6-n^2+7)}$$ converge or diverge? Justify.

I know that it diverges. I am trying to use the comparison test to prove it but I am having trouble finding a smaller series that diverges. Any help would be greatly appreciated.

Answer

HINT

Note that since $n^{\frac{1}{n}}\to 1$

$$\frac{2n^{5}+13n^{3}}{n^{\frac{1}{n}}(n^6-n^2+7)}\sim \frac2n$$

then use limit comparison test with $\sum \frac1n$.