# Cauchy Condensation Test

• Feb 29th 2008, 03:06 AM
disclaimer
Cauchy Condensation Test
Hi;

Can somebody please provide me with an example of using the Cauchy condensation test (criterion "$\displaystyle 2^k$") 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. :)

• Feb 29th 2008, 03:16 AM
mr fantastic
Quote:

Originally Posted by disclaimer
Hi;

Can somebody please provide me with an example of using the Cauchy condensation test (criterion "$\displaystyle 2^k$") 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. :)

I suppose that you already know that $\displaystyle \sum^\infty_{k=1}a_k$ is a convergent series iff $\displaystyle \sum^\infty_{k=1}2^ka_{2k}$ 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
$\displaystyle \sum_{k=1}^\infty\frac{1}{k}$ transforms to$\displaystyle \sum^\infty_{k=1}\frac{2^k}{2^k}=\sum_{k=1}^\infty 1$