Category:Dirichlet Beta Function at Odd Positive Integers

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This category contains pages concerning Dirichlet Beta Function at Odd Positive Integers:


\(\ds \map \beta {2 n + 1}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k} {\paren {2 k + 1}^{2 n + 1} }\)
\(\ds \) \(=\) \(\ds \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots\)
\(\ds \) \(=\) \(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\)

where:

$\beta$ denotes the Dirichlet beta function
$E_n$ denotes the $n$th Euler number
$n$ is a non-negative integer.
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