Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1/Proof 1

Theorem

$\map \beta 1 = \dfrac \pi 4 $

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

\(\ds \leadsto \ \ \)

\(\ds \map \beta 1\)

\(=\)

\(\ds \paren {-1}^0 \dfrac {E_0 \pi} {4 \paren 0!}\)

setting $n \gets 0$

\(\ds \)

\(=\)

\(\ds \frac \pi 4\)

Factorial of Zero, Zeroth Power of Real Number equals One and Euler Number Values:$E_0 = 1$

$\blacksquare$