My apologies. I couldn't get the /frac to work around the whole expression. Its supposed to be e^(1/n^9) / n^10.
It's not hard to check that $\displaystyle e^{\frac{1}{n^9}}\xrightarrow [n\to \infty] {} 1$, so you can use the comparison test for positives series: for n big enough, $\displaystyle \frac{e^{\frac{1}{n^9}}}{n^{10}}\leq \frac{2}{n^{10}}$...