Mercator series

Polynomial approximation to logarithm with n=1, 2, 3, and 10 in the interval (0,2).

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:

${\displaystyle \ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots }$

In summation notation,

${\displaystyle \ln (1+x)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}{x}^n.}$

The series converges to the natural logarithm (shifted by 1) whenever ${\displaystyle -1<x\le 1}$ .

History

The series was discovered independently by Johannes Hudde^[1] and Isaac Newton. It was first published by Nicholas Mercator, in his 1668 treatise Logarithmotechnia.

Derivation

The series can be obtained from Taylor's theorem, by inductively computing the n^th derivative of ${\displaystyle \ln (x)}$ at ${\displaystyle x=1}$ , starting with

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

Alternatively, one can start with the finite geometric series (${\displaystyle t\ne -1}$)

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

which gives

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

It follows that

${\displaystyle {\int }_0^x\frac{dt}{1+t}={\int }_0^x(1-t+t^2-\cdots +(-t{)}^{n-1}+\frac{(-t{)}^n}{1+t})dt}$

and by termwise integration,

${\displaystyle \ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +(-1{)}^{n-1}\frac{x^n}{n}+(-1{)}^n{\int }_0^x\frac{t^n}{1+t}dt.}$

If ${\displaystyle -1<x\le 1}$ , the remainder term tends to 0 as ${\displaystyle n\to \mathrm{\infty }}$.

This expression may be integrated iteratively k more times to yield

${\displaystyle -x{A}_k(x)+B_k(x)\ln (1+x)=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}\frac{x^{n+k}}{n(n+1)\cdots (n+k)},}$

where

${\displaystyle A_k(x)=\frac{1}{k!}\sum _{m=0}^k(\genfrac{}{}{0ex}{}{k}{m})x^m\sum _{l=1}^{k-m}\frac{(-x{)}^{l-1}}{l}}$

and

${\displaystyle B_k(x)=\frac{1}{k!}(1+x{)}^k}$

are polynomials in x.^[2]

Special cases

Setting ${\displaystyle x=1}$ in the Mercator series yields the alternating harmonic series

${\displaystyle \sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k}=\ln (2).}$

Complex series

The complex power series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n}=z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{4}+\cdots }$

is the Taylor series for ${\displaystyle -\log (1-z)}$ , where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number ${\displaystyle |z|\le 1,z\ne 1}$. In fact, as seen by the ratio test, it has radius of convergence equal to 1, therefore converges absolutely on every disk B(0, r) with radius r < 1. Moreover, it converges uniformly on every nibbled disk ${\displaystyle \overline{B(0,1)}\setminus B(1,\delta )}$, with δ > 0. This follows at once from the algebraic identity:

${\displaystyle (1-z)\sum _{n=1}^m\frac{z^n}{n}=z-\sum _{n=2}^m\frac{z^n}{n(n-1)}-\frac{z^{m+1}}{m},}$

observing that the right-hand side is uniformly convergent on the whole closed unit disk.

See also

• John Craig

References

• Weisstein, Eric W.. "Mercator Series". http://mathworld.wolfram.com/MercatorSeries.html. 
• Anton von Braunmühl (1903) Vorlesungen über Geschichte der Trigonometrie, Seite 134, via Internet Archive
• 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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