# Definition:Mercator's Constant

(Redirected from Natural Logarithm of 2)

## Definition

Mercator's constant is the real number:

 $\ds \ln 2$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n$ $\ds$ $=$ $\ds 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb$ $\ds$ $=$ $\ds 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots \ldots$

## Also known as

Mercator's constant is also known as the alternating harmonic series.

Some sources refer to it as Gregory's constant, either for James Gregory, or Grégoire de Saint-Vincent, both of whom were early pioneers into the result of sums of convergent series.

## Source of Name

This entry was named for Nicholas Mercator.