I'm attempting to truly understand the radius of convergence.
Say I have a series $$\sum_{n=0} a_{n} x^n$$ with radius of convergence R. How do I describe, in terms of R, the radius of convergence of i) $$\sum_{n=0} a_{n}^p x^n$$ ii) $$\sum_{n=0} a_{n} x^{np} $$ In both cases, p is a natural number.
Thank you for your help.
We have $ \lim \sup |a_n^p|^{1/n}= (\lim \sup |a_n|^{1/n})^p$, hence the power series $\sum_{n=0} a_{n}^p x^n$ has radius of convergence $R^p$.
The series $\sum_{n=0} a_{n} x^{np}$ converges for $|x|^p <R$ and diverges for $|x|^p >R$. Therefore $\sum_{n=0} a_{n} x^{np}$ has radius of convergence $R^{1/p}$.