I'm unsure which test to use for this series: $\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$? I want to determine if this series converges or diverges: $$\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$$
I tried the Ratio Test at first, and didn't get anywhere with that. I'm thinking I have to use Comparison Test, and compare the test to the series $\sum{\frac{1}{n^2}}$ for convergence? But I wasn't sure how to prove that in this case. Can someone help me out here?

You may write, as $n \to \infty$,
$$\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}= \frac{\sqrt{n}}{2n^2}\frac{e^{\large \frac{\ln 3}{n}}}{1-\frac{5}{2n^2}}=\frac{1}{2}\frac{1}{n^{3/2}}\frac{1+\frac{\ln 3}{n}+\mathcal{O}\left(\frac {1}{n^2}\right)}{1-\frac{5}{2n^2}}\sim \frac{1}{2}\frac{1}{n^{3/2}}$$ and your initial series is convergent as is the series $\displaystyle \sum\frac{1}{n^{3/2}}$.
