TITLE: I want to study $\sqrt[n]{n}$ and its behavior.
QUESTION [5 upvotes]: As I was studying some limit problems, I came across 
$$\sqrt[n]{n}$$ 
and astoundingly found out that the graph of this has a maximum when $n = e$.
I thought there is no way that this is not a famous fact and I am very interested in it. I looked up some words such as "nth roots" or "rational exponents" but I haven't found this fact right away.
Can someone guide me to a link or tell me at least what this expression goes by in order to do a little bit of researching ?

REPLY [6 votes]: Simply derive to find the maximum, but first note that
$$\sqrt[n]{n} =n^{1/n}=\exp(\log(n^{1/n}))$$
This should explain the connection with $e$. In more detail:
$$\frac{dn^{1/n}}{dn}=n^{1/n}\frac{1-\log n}{n^2}$$
So the derivative is only zero when $\log n = 1 \to \ n=e$.