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 Not publicly viewable

  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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