Integral test for convergence: $\sum _1^\infty \frac{e^{1/n}}{n^2}$ Integral test for convergence: $$\sum _1^\infty \frac{e^{1/n}}{n^2}$$
I tried approaching this as an IBP but I haven't been able to sort the solution. Can this be made into a improper integral? and if so could someone show me the process?

If you really want to do this with the integral test, we first need to realize that the function $\dfrac{e^{1/x}}{x^2}$ is decreasing (which it is, as it has negative derivative) and is positive (which is pretty clear).
Then we may use the integral test. We consider the integral
$$\int_1^\infty \dfrac{e^{1/x}}{x^2} \mathrm{d}x.$$
We do the substitution $u = \frac{1}{x}$ to see that this is the same as
$$\int_0^1 e^u \mathrm{d}u,$$
which is clearly finite. Thus the series converges. $\diamondsuit$
