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mathematical analysis > calculus > integral calculus > sophomore's dream

number > mathematical constant > sophomore's dream

Preferred term

sophomore's dream

Definition(s)

In mathematics, the sophomore's dream is the pair of identities (especially the first)
${\\displaystyle {\egin{alignedat}{2}&\\int _{0}^{1}x^{-x}\\,dx&&=\\sum _{n=1}^{\\infty }n^{-n}\\\\&\\int _{0}^{1}x^{x}\\,dx&&=\\sum _{n=1}^{\\infty }(-1)^{n+1}n^{-n}=-\\sum _{n=1}^{\\infty }(-n)^{-n}\\end{alignedat}}}$
discovered in 1697 by Johann Bernoulli.
The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively.
The name "sophomore's dream" is in contrast to the name "freshman's dream" which is given to the incorrect identity ${\ extstyle (x+y)^{n}=x^{n}+y^{n}}$ . The sophomore's dream has a similar too-good-to-be-true feel, but is true.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Sophomore%27s_dream)

Broader concept(s)

In other languages

rêve du deuxième année

French

URI

http://data.loterre.fr/ark:/67375/PSR-N164SZ6W-8

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