Mercator series
From Wikipedia, the free encyclopedia
In mathematics, the Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm. It is given by
valid for 
.
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[edit] History
The series was discovered independently by Isaac Newton, Nicholas Mercator and Gregory Saint-Vincent. It was first published by Mercator, in his 1668 treatise Logarithmo-technica.
[edit] Derivation
The series can be derived by repeatedly differentiating the natural logarithm, starting with
Alternatively, one can start with the geometric series (
)
which gives
It follows that
and by termwise integration,
If , the remainder term vanishes when 
.
[edit] Special cases
Setting x = 1, the Mercator series reduces to the alternating harmonic series
[edit] References
- Eric W. Weisstein, Mercator Series at MathWorld.
- Eriksson, Larsson & Wahde. Matematisk analys med tillämpningar, part 3. Gothenburg 2002. p. 10.
- Some Contemporaries of Descartes, Fermat, Pascal and Huygens from A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball
