Mercator Series
المؤلف:
Borwein, J. and Bailey, D
المصدر:
Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters
الجزء والصفحة:
...
24-6-2019
1900
Mercator Series
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.
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.
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