page_title: Sophomore's dream
section_title: Proof
hierarchical_section_title: Sophomore's dream / Proof
caption_attribution_description: Graph showing the functions featured in the Sophomore's Dream.
context_page_description: In mathematics, the sophomore's dream is the pair of identities
  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", which appears in, is in contrast to the name "freshman's dream" which is given to the incorrect identity ⁿ = xⁿ + yⁿ. The sophomore's dream has a similar too-good-to-be-true feel, but is true.
context_section_description: The proofs of the two identities are completely analogous, so only the proof of the second is presented here.
  The key ingredients of the proof are:
  to write xˣ = exp(x log x) (using the notation exp(t) for the exponential function eᵗ to base e);
  to expand exp(x log x) using the power series for exp; and
  to integrate termwise, using integration by substitution.
  In details, one expands xˣ as
  Therefore, 
  By uniform convergence of the power series, one may interchange summation and integration to yield
  To evaluate the above integrals, one may change the variable in the integral via the substitution   With this substitution, the bounds of integration are transformed to  giving the identity
  By Euler's integral identity for the Gamma function, one has  
  so that
  Summing these (and changing indexing so it starts at n = 1
  instead of n = 0) yields the formula.
caption: Graph of the functions y = xx (red, lower) and y = x−x (grey, upper) on the interval x ∈ (0, 1].
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