Let $R$ be radius of convergence for $\sum^{\infty}_{n=0} a_n z^n$. find radius of convergence for $\sum^{\infty}_{n=0} a_n z^{kn+l}$ in terms of $R$.

Question

Let $R \in [0,\infty)$ be radius of convergence for $$\sum^{\infty}_{n=0} a_n z^n$$. For $k \in \mathbb N, l \in \mathbb N_0$ find radius of convergence for $$\sum^{\infty}_{n=0} a_n z^{kn+l}$$ in terms of $R$.

I see that the coefficients are the same, but the power of $z$ corresponds to a term $kn+l$ positions further ahead in the original series. How can I formulate this in terms of $R$ ?

Answer 1

Let $A(z):=\sum_{n=0}^{\infty}a_nz^n$. Note that $z^l$ is multiplied to all the terms so that you can take it outside the summation and then the summation reads $$z^l\sum_{n=0}^{\infty}a_ny^n=z^lA(y)$$ where $y=z^k$ Since $R$ is the radius of convergence of the series denoted by $A(y)$, we need to have $|y|<R\Rightarrow |z|<R^{1/k}$.