Mercator series

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

\ln (1%2Bx) \;=\; x \,-\, \frac{x^2}{2} \,%2B\, \frac{x^3}{3} \,-\, \frac{x^4}{4} \,%2B\, \cdots.

In summation notation,

\ln (1%2Bx) \;=\; \sum_{n=1}^\infty \frac{(-1)^{n%2B1}}{n} x^n.

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

Contents

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.

Derivation

The series can be obtained from Taylor's theorem, by inductively computing the nth 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 %2B t^2 - \cdots %2B (-t)^{n-1} = \frac{1 - (-t)^n}{1%2Bt}

which gives

\frac{1}{1%2Bt} = 1 - t %2B t^2 - \cdots %2B (-t)^{n-1} %2B \frac{(-t)^n}{1%2Bt}.

It follows that

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

and by termwise integration,

\ln(1%2Bx) = x - \frac{x^2}{2} %2B \frac{x^3}{3} - \cdots %2B (-1)^{n-1}\frac{x^n}{n} %2B (-1)^n \int_0^x \frac{t^n}{1%2Bt} \,dt.

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

This expression may be integrated iteratively k more times to yield

-xA_k(x)%2BB_k(x) \ln (1%2Bx) = \sum_{n=1}^\infty (-1)^{n-1}\frac{x^{n%2Bk}}{n(n%2B1)\cdots (n%2Bk)},

where

A_k(x) = \frac{1}{k!}\sum_{m=0}^k{k \choose m}x^m\sum_{l=1}^{k-m}\frac{(-x)^{l-1}}{l}

and

B_k(x)=\frac{1}{k!}(1%2Bx)^k

are polynomials in x.[1]

Special cases

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

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

Complex series

The complex power series

z \,-\, \frac{z^2}{2} \,%2B\, \frac{z^3}{3} \,-\, \frac{z^4}{4} \,%2B\, \cdots

is the Taylor series for ln(1 + z), where ln denotes the principal branch of the complex logarithm. This series converges within the open unit disk |z| < 1 and on the circle |z| = 1 except at z = -1 (due to Abel's test), and the convergence is uniform on each closed disk of radius strictly less than 1.

References

  1. ^ Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. (2009). "Iterated primitives of logarithmic powers". arXiv:0911.1325.