Prove serial representation of the integral $\int_0^1 x^x \,{\rm d}x$

Question

I have to prove the serial representation of: \begin{equation*} \int^1_0 x^x\,{\rm d}x=\sum^{\infty}_{n=0}\frac{(-1)^{n-1}}{n^n}. \end{equation*}

It obtains: \begin{equation*} x^x=\sum^{\infty}_{n=0}\frac{(f(x))^n}{n!}. \end{equation*}

The function $f(x)$ is continuous in $[0,1]$.

http://math.stackexchange.com/questions/237513/series-as-an-integral-sophomores-dream — Jack D'Aurizio, May 23 '15 at 18:30

score 1 · Accepted Answer · answered May 23 '15 at 18:27

1

Have you ever heard of the Sophomore's Dream? In this wiki article there is a proof of the first identity. I believe this solves your problem.

answered May 23 '15 at 18:27

Ivo Terek

1 Answers1