Mercator series

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}^\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 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 + 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 ≤ 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)+B_k(x) \ln (1+x) = \sum_{n=1}^\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{k \choose 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.[1]

Special cases

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

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

Complex series

The complex power series

\sum_{n=1}^\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 \scriptstyle \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

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