Let $f$ be a strictly increasing positive continuous function defined on $[1,\infty)$ with limit $\infty$ as $x$ goes towards $\infty$. Then $\sum_{k=0}^{\infty} \frac{1}{f(k)}$ converges if and only if $\sum_{k=0}^{\infty} \frac{f^{-1}(k)}{k^2}$ converges, where $f^{-1}$ denotes the inverse of $f$.
I saw this claim as an exercise in a analysis textbook, linked from this site. Can not remember which one unfortunately. It was listed as a challenging exercise and has proven too challenging for me.
My initial idea was to try to use the integral test. $\sum \frac{1}{f(k)}$ converges exactly when $\int \frac{1}{f(x)}$ converges. I thought I might do some smart change of variable to find that $\int \frac{f^{-1}(x)}{x^2}$ converges. I could not come up with one however and I also realized that I do not know that the sum $\sum \frac{f^{-1}(k)}{k^2}$ satisfies the conditions for using the integral test. Unfortunately I ran out of ideas at that point.