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Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function

Dan Kalman and Mark McKinzie
The American Mathematical Monthly
Vol. 119, No. 1 (January 2012), pp. 42-51
DOI: 10.4169/amer.math.monthly.119.01.042
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.01.042
Page Count: 10
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Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function
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Abstract

Abstract It is tempting to try to reprove Euler’s famous result that \documentclass{article} \pagestyle{empty}\begin{document} $\sum 1/k^2 = \pi^2/6$ \end{document} using power series methods of the sort taught in calculus 2. This leads to \documentclass{article} \pagestyle{empty}\begin{document} $\int_0^1 -\frac{\ln(1-t)}{t} dt$ \end{document} , the evaluation of which presents an obstacle. With two key identities the obstacle is overcome, proving the desired result. And who discovered the requisite identities? Euler! Whether he knew of this proof remains to be discovered.

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