Cauchy Condensation Test
The Cauchy condensation test is a convergence test. It’s also, from what I can see, one of the few we didn’t cover in Analysis. Which is a shame, because it’s rather nice.
For a positive non-increasing sequence , the sum converges if and only if the sum converges.
A sketch proof in one direction should be rather evident: as the sequence is non-increasing, we can replace every group of length by its initial value. For the reverse, the idea is similar.
Let us consider our old friend . Consider which is a geometric series and convergent if and only if .