How do you construct a sequence of functions $f_n(x)$ such that
$s = \limsup_{n\rightarrow \infty} \sqrt[n]{f_n(x)}$
for all $s > 0$?
I know it's possible to this with a different sequence
$s = \limsup_{n\rightarrow \infty} (1 + \frac{x}{n})^n$
where $x = \log(s)$.
The motivation is from proofs on radius of convergence which rely on the definition of the radius
$r = \frac{1}{\limsup_{n\rightarrow \infty} \sqrt[n]{f_n(x)}}$
and out of curiosity I tried to construct a function similar to that for $e^x$ that could map to any $s = 1/r$ but couldn't.