Mercator series

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 ${\displaystyle -1<x\leq 1}$ .

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 n^th derivative of ${\displaystyle \ln(x)}$ at ${\displaystyle x=1}$ , starting with

...
Wikipedia