May 04, 2012

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 $f(n)$, the sum $\sum_{n=1}^{\infty}f(n)$ converges if and only if the sum $\sum_{n=1}^{\infty}2^nf(2^n)$ converges.

A sketch proof in one direction should be rather evident: as the sequence is non-increasing, we can replace every group of length $2^n$ by its initial value. For the reverse, the idea is similar.

Let us consider our old friend $\sum_{n=1}^{\infty}\frac{1}{n^p}$. Consider $\sum_{n=1}^{\infty}2^n(\frac{1}{2^n})^p=\sum_{n=1}^{\infty}2^{n-np}=\sum_{n=1}^{\infty}2^{n(1-p)}$ which is a geometric series and convergent if and only if $p>1$.

2 comments by 1 or more people

1. Nick

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

05 May 2012, 09:28

2. Kwok Tsoi

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

07 May 2012, 00:48

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