Mercator series

Short description: Taylor series for the natural logarithm

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:

[math]\displaystyle{ \ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots }[/math]

In summation notation,

[math]\displaystyle{ \ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. }[/math]

The series converges to the natural logarithm (shifted by 1) whenever [math]\displaystyle{ -1\lt x\le 1 }[/math] .

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 [math]\displaystyle{ \ln(x) }[/math] at [math]\displaystyle{ x=1 }[/math] , starting with

[math]\displaystyle{ \frac{d}{dx}\ln(x)=\frac1{x}. }[/math]

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

[math]\displaystyle{ 1-t+t^2-\cdots+(-t)^{n-1}=\frac{1-(-t)^n}{1+t} }[/math]

which gives

[math]\displaystyle{ \frac1{1+t}=1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}. }[/math]

It follows that

[math]\displaystyle{ \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 }[/math]

and by termwise integration,

[math]\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. }[/math]

If [math]\displaystyle{ -1\lt x\le 1 }[/math] , the remainder term tends to 0 as [math]\displaystyle{ n\to\infty }[/math].

This expression may be integrated iteratively k more times to yield

[math]\displaystyle{ -xA_k(x)+B_k(x)\ln(1+x)=\sum_{n=1}^\infty (-1)^{n-1}\frac{x^{n+k}}{n(n+1)\cdots (n+k)}, }[/math]

where

[math]\displaystyle{ A_k(x)=\frac1{k!}\sum_{m=0}^k{k\choose m}x^m\sum_{l=1}^{k-m}\frac{(-x)^{l-1}}{l} }[/math]

and

[math]\displaystyle{ B_k(x)=\frac1{k!}(1+x)^k }[/math]

are polynomials in x.^[2]

Special cases

Setting [math]\displaystyle{ x=1 }[/math] in the Mercator series yields the alternating harmonic series

[math]\displaystyle{ \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k}=\ln(2). }[/math]

Complex series

The complex power series

[math]\displaystyle{ \sum_{n=1}^\infty \frac{z^n}{n}=z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{4}+\cdots }[/math]

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

[math]\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}, }[/math]

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

References

↑ Vermij, Rienk (3 February 2012). "Bijdrage tot de bio-bibliografie van Johannes Hudde" (in nl). GEWINA / TGGNWT 18 (1): 25–35. ISSN 0928-303X. https://dspace.library.uu.nl/handle/1874/251283.
↑ Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. (2011). "Iterated primitives of logarithmic powers". International Journal of Number Theory 7 (3): 623–634. doi:10.1142/S179304211100423X.