Does the series from $1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge?

Question

Does the series from $n = 1$ to $\infty$ of $\frac{e^{1/n}}{n^2}$ converge or diverge? Steps/tips would be greatly appreciated. Thanks!

Answer 1

Converge. $\max_n e^{1/n}$ is bounded. $\sum_n n^{-2}$ converges.

Answer 2

As an alternate to using the fact that $\sum n^{-2} < \infty$, one can use the integral test:

$$\int_0^{\infty} \frac{e^{1/n}}{n^2} dn = -e^{1/n} \Big|_0^{\infty} = 1 < \infty$$