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