Let $R$ be the radius of convergence of the power series $\sum_{n=0}^{\infty}a_nz^n$, Then, the radius of convergence of $\sum_{n=0}^{\infty}a_{kn}z^n$ for a fixed positive integer $k$ is……?
If $a_n$ is non negative and converging, then by Cauchy Hadamard, radius of convergence is same, but what about the else?
Best Answer
It depends. For instance if$$a_n=\begin{cases}0&\text{ if $n$ is even}\\1&\text{ otherwise,}\end{cases}$$then $R=1$, but the radius of convergence of $\sum_{n=0}^\infty a_{2n}z^n$ is $\infty$.