Discuss the convergence or divergence of the series .
By now, you probably know that this series converges since it is a p-series with p=2. However, let us look at this series again using the Cauchy condensation test. By this test, we know that our series and the series
converge or diverge together. It is easy to show that this series converges by simplifying it.
This is just a geometric series with common ratio and first term which is well known to converge to 1. Thus, this series converges and therefore the original series converges by the Cauchy condensation test.