Mercator series

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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 \le 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.

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[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.

[edit] Derivation

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 \neq -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 \le 1, the remainder term vanishes when n \to \infty.

[edit] Special cases

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

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

[edit] References