If I have the following sum $$ \sum_{k\geq1} \frac{1}{r^2_k}, $$ is it possible that this series converges? The sum of $1/n, n \in \mathbb{N}$ diverges but in my case it's about real numbers and not natural.
If $r_k$ grows very fast, then this sum will get small very fast, so is it correct to say that it will converge, i.e. $< \infty$?
Sorry for low effort question but I'm a bit lost. Any insight would be great, thanks.
EDIT: I'm reading a paper non-related to calculus but it has this sum and I think the author implies that this sum converges when $r_k \to \infty$ sufficiently fast. Is that correct way of thinking or it's wrong?
It converges if $r_k >k$ for all $k$ because $\sum \frac 1 {k^{2}} <\infty$.