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 Johannes Hudde and Isaac Newton. It was first published by Nicholas 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.