$\lim_{n\to\infty} \sqrt[n]{n!}$ and $\lim_{n\to\infty}\frac{1}{n} \sqrt[n]{n!}$ with differentiation and integration tools.

real-analysis limits riemann-integration stieltjes-integral

Solution 1:

I'd suggest you start with the second problem by taking the logarithm and simplifying; the key simplification you need is $\log(n!)=\sum_{k=1}^n \log(k)$. Once you understand the second one, the first one is straightforward.

Incidentally this trick of starting off with the logarithm is the usual procedure for studying infinite products and similar expressions.

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