False Logarithm Series

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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 (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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