For example harmonic series corresponds to Cauchy condensed series of $1+1+1+\cdots$ and since the Cauchy condensation is exponential, it just seems natural to reverse of it being related to order of convergence/divergence, which as it happens in the case of harmonic series the order of divergence is $\ln n$.
Are there cases were there are modified Cauchy condensation can be applied, for some fuction $f$, where $f$ is a monotonically increasing function greater than 1? i.e. something like $\sum \frac{f(n)}{a_{f(n)}}$? would there be cases where the convergence of standard Cauchy condensed series is just as hard to decide as the original series but a bigger condensation like $2^{2^n}$ or $n!$ would be easy/immediately verifiable?
