Sophomore's Dream

Theorem

Sophomore's Dream refers to two identities discovered in 1697 by Johann Bernoulli.

$\ds \int_0^1 x^x \rd x$

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n}$

$\ds $

$=$

$\ds -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n}$

$\ds $

$=$

$\ds 0.78343 \ 05107 \ 12 \ldots$

$\ds \int_0^1 x^{-x} \rd x$

$=$

$\ds \sum_{n \mathop = 1}^\infty n^{-n}$

$\ds $

$=$

$\ds 1.29128 \ 5997 \ldots$