Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5
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Theorem
Let $\beta$ denote the Dirichlet beta function.
Then:
- $\map \beta 5 = \dfrac {5 \pi^5} {1536} $
Proof
\(\ds \map \beta {2 n + 1}\) | \(=\) | \(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\) | Dirichlet Beta Function at Odd Positive Integers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \beta 5\) | \(=\) | \(\ds \paren {-1}^2 \dfrac {E_4 \pi^5 } {4^3 \paren {4}!}\) | setting $n := 2$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {5 \pi^5} {1536}\) | Euler Number Values: $E_4 = 5$ |
$\blacksquare$