# On the radius of convergence of a power series

by piggees
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,348 On the radius of convergence of a power series It is, basically, an application of the "ratio test". If $f(z)= \sum a_n(z- z_0)^n$] then the series converges, absolutely, as long as $$\lim_{n\to\infty}\frac{|a_{n+1}(z- z_0)^{n+1}|}{|a_n (z- z_0)^n|}$$$$= |z- z_0|\lim_{n\to\infty}\frac{a_{n+1}{a_n}|< 1$$ and diverges if that limit is larger than 1. As long as $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}= A$$ exists, then we have that the power series converges for $$|z- z_0|< \frac{1}{A}$$ and diverges for $$|z- z_0|> \frac{1}{A}$$ You can get the same result by using the root test instead of the ratio test: $\sum a_n (z- z_0)^n$ converges absolutely as long as $$\lim_{n\to\infty}\left(a_n(z- z_0)^n)^{1/n}= \left(\lim_{n\to\infty}\sqrt[n]{a_n}\right)|z- z_0|$$ is less than 1.