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MERCATOR SERIES

In mathematics, the 'Mercator series' or 'Newton-Mercator series' is the Taylor series for the natural logarithm. It is given by
:ln(1+x)=sum_{n=1}^infty rac{(-1)^{n+1}}{n} x^n = x - rac{x^2}{2} + rac{x^3}{3} - rac{x^4}{4} + cdots,
valid for -1 < x le 1.

Contents
History
Derivation
Special cases
References

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''.

Derivation


The series can be derived by repeatedly differentiating the natural logarithm, starting with
: rac{d}{dx} ln x = rac{1}{x}.
Alternatively, one can start with the geometric series (t
eq -1)
:1 - t + t^2 - cdots + (-t)^{n-1} = rac{1 - (-t)^n}{1+t}
which gives
: rac{1}{1+t} = 1 - t + t^2 - cdots + (-t)^{n-1} + rac{(-t)^n}{1+t}.
It follows that
:int_0^x rac{dt}{1+t} = int_0^x left( 1 - t + t^2 - cdots + (-t)^{n-1} + rac{(-t)^n}{1+t}
ight) dt
and by termwise integration,
:ln(1+x) = x - rac{x^2}{2} + rac{x^3}{3} - cdots + (-1)^{n-1} rac{x^n}{n} + (-1)^n int_0^x rac{t^n}{1+t} dt.
If -1 < x le 1, the remainder term vanishes when n o infty.

Special cases


Setting x = 1, the Mercator series reduces to the alternating harmonic series
:sum_{k = 1}^infty rac{(-1)^{k + 1}}{k} = ln 2.

References



★

★ 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

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