Mercator Series

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

			(1)
			(2)

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.

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.