### 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 .

## 2 comments by 1 or more people

## Nick

A useful addition to the toolbox of tests – nice one!

05 May 2012, 09:28

## Kwok Tsoi

In reality,

we judge convergence of most series by “inspection” =p.

07 May 2012, 00:48

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