The series converges to the natural logarithm (shifted by 1) whenever −1 < x ≤ 1.
The series can be obtained from Taylor's theorem, by inductively computing the nth derivative of ln x at x = 1, starting with
Alternatively, one can start with the finite geometric series (t ≠ −1)
It follows that
and by termwise integration,
If −1 < x ≤ 1, the remainder term tends to 0 as .
This expression may be integrated iteratively k more times to yield
are polynomials in x.
Setting x = 1 in the Mercator series yields the alternating harmonic series
The complex power series
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 , 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.
- 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