TITLE: Convergence of a sequence with lots of powers
QUESTION [6 upvotes]: A problem from "Problems in Real Analysis - Advanced Calculus on the Real Axis":

Let $p$ be a nonnegative real number. Study the convergence of the sequence $(x_n)_{n\ge1}$ defined by $$x_n=\left(1^{1^p}\cdot2^{2^p}\cdots(n+1)^{(n+1)^p}\right)^{1/(n+1)^{p+1}}-\left(1^{1^p}\cdot2^{2^p}\cdots n^{n^p}\right)^{1/n^{p+1}}.$$

If $p=0$, it is already proved in the book that $x_n$ converges to $\dfrac1e$. So just consider the case that $p>0$. If the sequence converges, I can prove by Stolz-Cesàro Theorem that the sequence must converge to $0$. But I don't know how to determine which $p$ gives convergent sequences.
Please don't give complete solution, just helpful hints. Thanks in advance.

REPLY [3 votes]: HINT : Let $ a_n = \left( 1^{1^p} \cdot 2^{2^p} \cdots n^{n^p} \right)^{1/n^{p+1}}$ so that $x_n = a_{n+1}- a_n $. Now study the asymptotics of $\log a_n$ using the Euler-Maclaurin formula.