
Cauchy condensation test
I need to determine, using the Cauchy Condensation Test, whether or not
the series 1/(n * Log(n)) converges. I believe that this series converges iff
2^n(1/(2^n*Log(2^n)) converges and I believe that it actually diverges. But I am not sure how to work through it. Thanks for your help.

It looks like you already have it! You wrote down $\displaystyle \displaystyle \sum_{n=1}^\infty 2^n\frac{1}{2^n \log (2^n)}$. This is precisely $\displaystyle \displaystyle \sum_{n=1}^\infty \frac{1}{\log (2^n)}=\sum_{n=1}^\infty \frac{1}{n \log 2}=\displaystyle \frac{1}{\log 2}\sum_{n=1}^\infty \frac{1}{n}$, which diverges.