Let $f(x)$ be a positive function such that $\displaystyle\lim_{n\to \infty} \frac{f(n)}{n}=a>0.$
The question is how to calculate the following limit:
$$\lim_{n\to \infty}\sqrt[n+1]{\prod_{k=1}^{n+1} f(k)}-\sqrt[n]{\prod_{k=1}^n f(k)}.$$
I think it can be computed by Stolz Theorem. But I do not know how.
Any help will be appreciated.