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A nice identity concerning an integral and a serie

Source: Brazilian Math Olympiad 2005, Problem 5

October 24, 2005
calculusintegrationlogarithmsfunctioninductionreal analysisreal analysis solved

Problem Statement

Prove that ∑n=1∞1nn=∫01x−x dx. \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx.
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