0
$\begingroup$

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 $R$ ?

$\endgroup$
2
$\begingroup$

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 $|y|<R\Rightarrow |z|<R^{1/k}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.