The radius of convergence of $\sum_{n=0}^{\infty}a_{kn}z^n$ for a fixed positive integer $k$.

Question

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?

Answer 1

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 $\infty$.

Answer 2

What you can say is that $$ \limsup\sqrt[n]{|a_{kn}|}= \left(\limsup\sqrt[kn]{|a_{kn}|}\right)^k\le \left(\limsup\sqrt[n]{|a_{n}|}\right)^k$$ and hence $$ R_k\ge R^k.$$

The ineqquality need not be sharp in any way. As witnessed by letting $a_n=n^n$ if $n$ is prime and $a_n=0$ otherwise, it is possible that $R=0$ and $R_k=\infty$ for all $k>1$.