In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:
In summation notation,
The series converges to the natural logarithm (shifted by 1) whenever .
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.
The series can be obtained from Taylor's theorem, by inductively computing the nth derivative of at , starting with
Alternatively, one can start with the finite geometric series ()
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
are polynomials in x.
Setting in the Mercator series yields the alternating harmonic 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.