Definition:Catalan's Constant
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Definition
Catalan's constant is the real number defined as:
\(\ds G\) | \(=\) | \(\ds \map \beta 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^{\infty} \frac{\paren {-1}^n} {\paren {2 n + 1}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1^2} - \frac 1 {3^2} + \frac 1 {5^2} - \frac 1 {7^2} + \cdots\) |
where $\beta$ is the Dirichlet beta function.
Its numerical value is approximately:
- $G = 0 \cdotp 91596 \, 55941 \, 77219 \, 01505 \, 46035 \, 14932 \, 38411 \, 0774 \ldots$
This sequence is A006752 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Source of Name
This entry was named for Eugène Charles Catalan.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,91596 55941 77219 01505 \ldots$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Catalan's constant