Weierstrass M-test and Sophomore's Dream.

Question

I am trying to prove the Sophomore's Dream $\int_0^1 x^{-x} \ d x = \sum\limits_{n = 1}^\infty n^{-n}$. I get to this point, $\lim\limits_{a \to 0^+} \int_a ^1 \sum\limits_{n = 0}^\infty \frac{(-1)^n(x \log x)^n}{n!} dx$. But in order to interchange the sum and the integral, I need to show that $\sum\limits_{n = 0}^\infty \frac{(-1)^n(x \log x)^n}{n!}$ converges uniformly. I tried using the M-test, but I don't know how to get bounds on the $\big |\frac{(-1)^n(x \log x)^n}{n!} \big | $. Any help would be appreciated.

Answer 1

Hint: $|x\log x|\le 1$ for all $x\in(0,1)$.