I'm trying to teach myself complex analysis, and I've been working on this idea.
Suppose $\lim_{n\to\infty}|a_n|/|a_{n+1}|=R$, I would like to know why $\sum a_nz^n$ also has $R$ as its radius of convergence.
I believe I want to show $ \limsup \sqrt[n]{|a_n|}=\frac{1}{\lim_{n\to\infty}|a_n|/|a_{n+1}|}=\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}. $
I notice that $|a_n|$ can be written as $|a_n|=\frac{|a_n|}{|a_{n-1}|}\frac{|a_{n-1}|}{|a_{n-2}|}\cdots\frac{|a_{k+1}|}{|a_{k}|}|a_k|$. I hoped this would be helpful since it is a product of ratios of the absolute values of successive terms.
I didn't know how to proceed after these observations. How can I finish this? Thanks!