When I was solving an exercise related to power series with complex numbers, I had the next question. I thought that was trivial, but, in fact, it isn't.
Let $\displaystyle\sum_{n=0}^{\infty} a_nz^n$ a power series (centered at 0), where for all $n\in\mathbb{N}$, $a_n\in\mathbb{C}$, such that the series converges if and only if $|z|<R$, i.e., $R$ is the radius of convergence of the series. If we take $\{a_{n_k}\}_{k\in\mathbb{N}}\subseteq\{a_n\}_{n\in\mathbb{N}}$ a subsequence of $\{a_n\}$ then, what can we say about the radius of convergence $r$ of the series $\displaystyle\sum_{k\in\mathbb{N}}^{} a_{n_k}z^{n_k}$? Is $r$ related to $R$?
My first approach was thought that the radius was the same, but, it isn't true. Moreover, maybe the series doesn't converges for a certain values. Now I don't know how to proceed. Any hint?