Mercator series

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm. It is given by

$\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots,$

valid for −1 < x ≤ 1. For complex values of x, it converges uniformly to the principal branch of the complex logarithm for all x in the open unit disc.

[edit] History

The series was discovered independently by Nicholas Mercator, Isaac Newton and Gregory Saint-Vincent. It was first published by Mercator, in his 1668 treatise Logarithmo-technica.

The series can be derived by repeatedly differentiating the natural logarithm, starting with

$\frac{d}{dx} \ln x = \frac{1}{x}.$

Alternatively, one can start with the geometric series (t ≠ −1)

$1 - t + t^2 - \cdots + (-t)^{n-1} = \frac{1 - (-t)^n}{1+t}$

which gives

$\frac{1}{1+t} = 1 - t + t^2 - \cdots + (-t)^{n-1} + \frac{(-t)^n}{1+t}.$

It follows that

$\int_0^x \frac{dt}{1+t} = \int_0^x \left( 1 - t + t^2 - \cdots + (-t)^{n-1} + \frac{(-t)^n}{1+t} \right)\, dt$

and by termwise integration,

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1}\frac{x^n}{n} + (-1)^n \int_0^x \frac{t^n}{1+t} \,dt.$

If −1 < x ≤ 1, the remainder term tends to 0 as $n \to \infty$ .

Setting x = 1, the Mercator series reduces to the alternating harmonic series

$\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = \ln 2.$

Weistein, Eric W., "Mercator Series" from 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