I'm looking for a nice (full) proof of some convergence test,
Including...
$\displaystyle \Sigma a_k$ and $\displaystyle \Sigma 2^ka_{2^k}$
Cauchy Condensation Test
Tonio
I'd also like to point out that Cauchy Condensation Test is a generalization of Oresme's proof of the divergence of the harmonic series. It might be useful to have a look.
Harmonic series (mathematics) - Wikipedia, the free encyclopedia