False Logarithm Series

In a paper titled "Consideration of Some Series Which Are Distinguished by Special Properties", published in 1753, Leonhard Euler considers . He proves that for non-negative integers , just like , but shows that this function is different from . This Demonstration compares the plots of and for different values of . Red points correspond to pairs of integers on the line .


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The series discovered by Euler is an instance of a -series. In fact where is the -polygamma function. It was the finiteness of that Euler used to demonstrate that is indeed different from .
More information is available in Ed Sandifer's column "How Euler Did It" and in the original paper by Euler.
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