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?

  • 6
    $\begingroup$ Sophomore's dream $\endgroup$ – Daniel Fischer Nov 9 '16 at 20:11
  • 2
    $\begingroup$ Write $x^x = e^{x\ln x} = \sum_{n=0}^\infty (x\ln x)^n/n!$ and away you go ... $\endgroup$ – zhw. Nov 9 '16 at 20:11
  • $\begingroup$ In THIS ANSWER, I developed a generalized version of the expression of interest (i.e., Generalized Sophomore's Dream). $\endgroup$ – Mark Viola Nov 9 '16 at 21:17

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