Test the following series for convergence or divergence. Justify the decision.

The series converges. We apply the integral test (Theorem 10.11 on page 397 of Apostol), letting

To evaluate the integral we use integration by parts with

Then we have

Therefore, by the integral test we have

Hence, the sequence converges; therefore, converges which implies the convergence of

Mistake in the integration: works out the same though. Comparison with 1/n^2 also works.