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

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?

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 $$\infty$$.
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 $$k>1$$.