Radius of convergence of a power series - how can I be sure $\lim \frac{a_{n+1}}{a_n}$ exists? Let $\sum a_n x^n$ be a power series whose radius of convergence is $0<R<\infty$.
What is the radius of convergence, $R'$ of $\sum \frac {n^n}{n!} a_n x^n$?
If I know $\lim_{n\rightarrow\infty} \frac {a_{n+1}}{a_n}$ exists. this would be an easy problem, since (after simplification):
$\frac1{R'}=\lim_{n\rightarrow\infty}(\frac{1+n}n)^n\cdot\frac{a_{n+1}}{a_n}=\frac e R$
But what bothers me is, how can I be sure $\lim \frac{a_{n+1}}{a_n}$ exists in the first place?

edit: $\limsup$ is not enough. For example, in the series $ f_n(x) =
\begin{cases}
\frac xn,  & \text{if $n$ is even} \\
\frac {2x}n, & \text{if $n$ is odd}
\end{cases}$
The radius of convergence is 1 but $\limsup \frac {a_{n+1}}{a_n}=2$.

Hint: Convergence is not necessary; consider $\limsup$.
