# Mercator series

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

The series converges to the natural logarithm (shifted by 1) whenever .

## 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 at , starting with

Alternatively, one can start with the finite geometric series ()

which gives

It follows that

and by termwise integration,

If , the remainder term tends to 0 as .

This expression may be integrated iteratively *k* more times to yield

where

and

are polynomials in *x*.[1]

## Complex series

The complex power series

is the Taylor series for , where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number . 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 , with *δ* > 0. This follows at once from the algebraic identity:

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

## References

- Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. (2009). "Iterated primitives of logarithmic powers".
*International Journal of Number Theory*.**7**: 623–634. arXiv:0911.1325. doi:10.1142/S179304211100423X.

- Weisstein, Eric W. "Mercator Series".
*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