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 Logarithmo-technica.

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

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

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.

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

$z \,-\, \frac{z^2}{2} \,+\, \frac{z^3}{3} \,-\, \frac{z^4}{4} \,+\, \cdots$

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

• Weisstein, Eric W., "Mercator Series" from MathWorld.
• 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