Chiral Potts curve

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

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

In summation notation,

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

The series converges to the natural logarithm (shifted by 1) whenever −1 < x ≤ 1.

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 Logarithmotechnia.

Derivation

The series can be obtained from Taylor's theorem, by inductively computing the n^th derivative of ln x at x = 1, starting with

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

Alternatively, one can start with the finite 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

$\underset{0}{\overset{x}{\int }}\frac{dt}{1+t}=\underset{0}{\overset{x}{\int }}(1-t+t^2-\cdots +(-t{)}^{n-1}+\frac{(-t{)}^n}{1+t})dt$

and by termwise integration,

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +(-1{)}^{n-1}\frac{x^n}{n}+(-1{)}^n\underset{0}{\overset{x}{\int }}\frac{t^n}{1+t}dt.$

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

This expression may be integrated iteratively k more times to yield

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

where

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

and

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

are polynomials in x.

Special cases

Setting x = 1 in the Mercator series yields the alternating harmonic series

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

Complex series

The complex power series

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

is the Taylor series for -log(1 - z), where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number |z| ≤ 1, z ≠ 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 $\overline{B(0,1)}\setminus B(1,\delta )$, with δ > 0. This follows at once from the algebraic identity:

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

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

References

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• 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