# Definition:Newton-Mercator Series

## Definition

Let $\ln x$ denote the natural logarithm function.

Then:

 $\ds \map \ln {1 + x}$ $=$ $\ds x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n$

The series converges to the natural logarithm (shifted by $1$) for $-1 < x \le 1$.

This is known as the Newton-Mercator series.

## Also known as

The Newton-Mercator series is also known as the Mercator series, or Mercator's series.

## Examples

### Newton-Mercator Series: $\ln 2$

The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$:

 $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n$ $=$ $\ds 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb$ $\ds$ $=$ $\ds \ln 2$

This real number is known as Mercator's constant.

## Also see

• Results about Newton-Mercator series can be found here.

## Source of Name

This entry was named for Isaac Newton and Nicholas Mercator.

## Historical Note

The Newton-Mercator Series was discovered independently by both Isaac Newton and Nicholas Mercator.

However, it was also independently discovered by GrĂ©goire de Saint-Vincent.