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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 .


	Contributed by: Oleksandr Pavlyk
	Show Initialization Code

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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