Sophomore's dream

In mathematics, sophomore's dream is a name occasionally used for the identities

$\sum_{n=1}^\infty n^{-n} = \int_0^1 x^{-x}\, dx\quad\quad(=1.291285997\dots)$

$\sum_{n=1}^\infty -(-1)^nn^{-n} = \int_0^1 x^{x}\, dx\quad\quad(= 0.783430510712\dots)$

discovered in 1697 by Johann Bernoulli.

The second identity can be proved by expanding x^x as

$\sum_{n=0}^\infty x^n\log(x)^n/n!$

and integrating term by term, and the first can be proved in a similar way.

[edit] References

Jonathan Borwein, David H. Bailey, Roland Girgensohn Experimentation in Mathematics: Computational Paths to Discovery Page 44.
William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, p. 46-51.
N. J. A. Sloane,(sequence A083648 in OEIS) and (sequence A073009 in OEIS)
Pólya and Gábor Szegö, Problems and Theorems in Analysis (part I, problem 160).
Weisstein, Eric W. Sophomore's Dream. From MathWorld--A Wolfram Web Resource.