proving pointwise convergence of a sequence

Question

I have a series (e.g. $\sum_{n=1}^\infty r^k $) I want to prove converges. If $\sum_{n=1}^\infty r^{k^2} \leq \sum_{n=1}^\infty r^k $, If I show the $\sum_{n=1}^\infty r^k $ converges, does it mean the $\sum_{n=1}^\infty r^{k^2}$ also converges?

Answer 1

$\sum_{n=1}^\infty r^k = \infty * r^k$ . It diverges..

If you mean $\sum_{n=1}^\infty r^n$ converges, then $|r| \lt 1$, therefore $|r^{n^2}| \lt |r^n|$ for $n>1$. so we have $0\le \sum_{n=1}^\infty |r^{n^2}|\le \sum_{n=1}^\infty |r^n|$. and $ \sum_{n=1}^\infty |r^n|$ converges since $|r| <1$, so is $\sum_{n=1}^\infty |r^{n^2}|$