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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): Table $1.1$. Mathematical Constants
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.18$
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,69314 7805 \ldots$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 693 \, 147 \, 180 \, 559 \, 945 \, 309 \, 417 \, 232 \, 121 \, 458 \, 176 \, 568 \, 075 \, 500 \, 134 \, 360 \ldots$
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Example $4$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots$