TITLE: Two Formulas for Radius of Convergence of power series.
QUESTION [5 upvotes]: Since $\limsup[(a_k)^{1/k}]$ is less than or equal to $\limsup[\frac{a_{k+1}}{a_k}]$ when $a_k$ is positive, how are the equations generated from the reciprocals of these expressions both the radius of convergence of a power series with coefficients $a_k$?

REPLY [8 votes]: They do not always give the same result.  The radius of convergence $R$ of a power series satisfies
$$
1/R \;=\; \limsup_{k\to\infty}\, (a_k)^{1/k}
$$
The other expression does not always give the radius of convergence.  For example, if $\{a_n\}$ is the sequence
$$
a_n = \begin{cases}2^n & \text{if }n\text{ is even} \\ 2^{n-1} & \text{if }n\text{ is odd,}\end{cases}
$$
then the ratio $a_{k+1}/a_k$ alternates between $1$ and $4$.   Thus $\displaystyle\limsup_{k\to\infty}\frac{a_{k+1}}{a_k} = 4$, even though the radius of convergence is $1/2$.