Sophomore's Dream
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High Quality Content by WIKIPEDIA articles! In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first) begin{align} int_0^1 x^{-x},dx &= sum_{n=1}^infty n^{-n}&&(= 1.291285997dots) int_0^1 x^x ,dx &= sum_{n=1}^infty (-1)^{n+1}n^{-n} = - sum_{n=1}^infty (-n)^{-n} &&(= 0.783430510712dots) end{align} discovered in 1697 by Johann Bernoulli. The name is in contrast to the "freshman's dream" which is given to the incorrect equation (x + y)n = xn + yn. (The correct result is given by the binomial theorem.) The sophomore's dream has a similar too-good-to-be-true feel, but is in fact true.…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first) begin{align} int_0^1 x^{-x},dx &= sum_{n=1}^infty n^{-n}&&(= 1.291285997dots) int_0^1 x^x ,dx &= sum_{n=1}^infty (-1)^{n+1}n^{-n} = - sum_{n=1}^infty (-n)^{-n} &&(= 0.783430510712dots) end{align} discovered in 1697 by Johann Bernoulli. The name is in contrast to the "freshman's dream" which is given to the incorrect equation (x + y)n = xn + yn. (The correct result is given by the binomial theorem.) The sophomore's dream has a similar too-good-to-be-true feel, but is in fact true.