Definition:Mercator's Constant

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Definition

Mercator's constant is the real number:

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

This sequence is A002162 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


Also see


Source of Name

This entry was named for Nicholas Mercator.


Sources