Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 3
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Theorem
Let $\beta$ denote the Dirichlet beta function.
Then:
- $\map \beta 3 = \dfrac {\pi^3} {32} $
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 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \beta 3\) | \(=\) | \(\ds \paren {-1}^1 \dfrac {E_2 \pi^3 } {4^2 \paren {2}!}\) | setting $n := 1$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^3} {32}\) | Euler Number Values: $E_2 = -1$ |
$\blacksquare$