ask the radius of convergence of power series
Baby Rudin 3.39 tells me the given the power series $\Sigma c_n z^n$, put
$$\alpha = \limsup \sqrt[n]{|c_n|}, R=1/\alpha$$
Then R is called the radius of convergence of $\Sigma c_n z^n$. But when I try to do exercise 3.9, the solution just uses $\lim \frac{a_n}{a_{n+1}}$ to find R. Why we can do this?
Solution 1:
There are various approches to calculate the radius of convergence:
- the one you use is based on the Root Test
- the one proposed is based on the Ratio Test
Both are valid and choosing one over the other depends on the given power series.