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 .
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 .