# Math Help - Cauchy Condensation Test

1. ## Cauchy Condensation Test

Hi;

Can somebody please provide me with an example of using the Cauchy condensation test (criterion " $2^k$") in checking convergence of series? I've never used it myself, and I'd really like to see a real example of application (it seems rather rare to use this criterion), not just the rule. I'd like to know how it works in practice.

2. Originally Posted by disclaimer
Hi;

Can somebody please provide me with an example of using the Cauchy condensation test (criterion " $2^k$") in checking convergence of series? I've never used it myself, and I'd really like to see a real example of application (it seems rather rare to use this criterion), not just the rule. I'd like to know how it works in practice.

3. I suppose that you already know that $\sum^\infty_{k=1}a_k$ is a convergent series iff $\sum^\infty_{k=1}2^ka_{2k}$ is a convergent series. The Cauchy Condensation test can mainly be used to prove divergence for various series and we can firstly use this to prove the divergence of the harmonic series
$\sum_{k=1}^\infty\frac{1}{k}$ transforms to $\sum^\infty_{k=1}\frac{2^k}{2^k}=\sum_{k=1}^\infty 1$