Sophomore's dream
In mathematics, the sophomore's dream is the pair of identities (especially the first) ∫ 0 1 x − x d x = ∑ n = 1 ∞ n − n ∫ 0 1 x x d x = ∑ n = 1 ∞ ( − 1 ) n + 1 n − n = − ∑ n = 1 ∞ ( − n ) − n {\displaystyle {\begin{alignedat}{2}&\int _{0}^{1}x^{-x}\,dx&&=\sum _{n=1}^{\infty }n^{-n}\\&\int _{0}^{1}x^{x}\,dx&&=\sum _{n=1}^{\infty }(-1)^{n+1}n^{-n}=-\sum _{n=1}^{\infty }(-n)^{-n}\end{alignedat}}} discovered in 1697 by Johann Bernoulli. The numerical values of these constants are approximately 1.291285997...