TOPICS
Search

Mercator Series


The Mercator series, also called the Newton-Mercator series (Havil 2003, p. 33), is the Taylor series for the natural logarithm

ln(1+x)=sum_(k=1)^(infty)((-1)^(k+1))/kx^k
(1)
=x-1/2x^2+1/3x^3-...
(2)

for -1<x<=1, which was found by Newton, but independently discovered and first published by Mercator in 1668.

Plugging in x=1 gives a beautiful series for the natural logarithm of 2,

 ln2=sum_(k=1)^infty((-1)^(k+1))/k,
(3)

also known as the alternating harmonic series and equal to eta(1), where eta(z) is the Dirichlet eta function.


See also

Alternating Harmonic Series, Dirichlet Eta Function, Logarithmic Number, Natural Logarithm, Natural Logarithm of 2

Explore with Wolfram|Alpha

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