# Equality of definite integral and infinite sum

The following appear, by numerical evaluation, to be equal

$\int_\limits{0}^1 x^x \, dx=\sum _\limits{n=1}^{\infty } (-1)^{n+1} n^{-n}$

Can anyone prove it?

• Sophomore's dream – Daniel Fischer Nov 9 '16 at 20:11
• Write $x^x = e^{x\ln x} = \sum_{n=0}^\infty (x\ln x)^n/n!$ and away you go ... – zhw. Nov 9 '16 at 20:11
• In THIS ANSWER, I developed a generalized version of the expression of interest (i.e., Generalized Sophomore's Dream). – Mark Viola Nov 9 '16 at 21:17