# Definition:Newton-Mercator Series

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## 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

- Power Series Expansion for Logarithm of 1 + x for a proof of its convergence

- 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.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Mercator's series**