The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural
logarithm
for 
,
which was found by Newton, but independently discovered and first published by Mercator
in 1668.
Plugging in 
gives a beautiful series for the natural logarithm
of 2,
 |
(3)
|
also known as the alternating harmonic series and equal to 
,
where 
is the Dirichlet eta function.
See also
Alternating Harmonic Series,
Dirichlet Eta Function,
Logarithmic
Number,
Natural Logarithm,
Natural
Logarithm of 2
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References
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, p. 49, 2003.Borwein, J. M.; Borwein, P. B.;
and Dilcher, K. "Pi, Euler Numbers, and Asymptotic Expansions." Amer.
Math. Monthly 96, 681-687, 1989.Havil, J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 14
and 33, 2003.Referenced on Wolfram|Alpha
Mercator Series
Cite this as:
Weisstein, Eric W. "Mercator Series."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MercatorSeries.html
Subject classifications
