The Sophomore's Dream

Manuel Eberl 🌐

April 10, 2022

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

This article provides a brief formalisation of the two equations known as the Sophomore's Dream, first discovered by Johann Bernoulli in 1697:

\[\int_0^1 x^{-x}\,\text{d}x = \sum_{n=1}^\infty n^{-n} \quad\text{and}\quad \int_0^1 x^x\,\text{d}x = -\sum_{n=1}^\infty (-n)^{-n}\]
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Topics

Theories of Sophomores_Dream