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

Theorem

Let $\beta$ denote the Dirichlet beta function.

Then:

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

Proof 1

\(\ds \map \beta {2 n + 1}\)

\(=\)

\(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\)

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:$E_0 = 1$

$\blacksquare$

Proof 2

\(\ds \frac 1 {1 + x^2}\)

\(=\)

\(\ds 1 - x^2 + x^4 - x^6 + \cdots\)

Sum of Infinite Geometric Sequence

\(\ds \int_0^1 \frac 1 {1 + x^2} \rd x\)

\(=\)

\(\ds \int_0^1 \paren {1 - x^2 + x^4 - x^6 + \cdots } \rd x\)

integrating both sides from $0$ to $1$

\(\ds \leadsto \ \ \)

\(\ds \map \arctan 1 - \map \arctan 0\)

\(=\)

\(\ds \intlimits {x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots } 0 1\)

Derivative of Arctangent Function, Primitive of Power

\(\ds \leadsto \ \ \)

\(\ds \frac \pi 4 - 0\)

\(=\)

\(\ds 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots\)

Arctangent of One, Arctangent of Zero is Zero

\(\ds \leadsto \ \ \)

\(\ds \frac \pi 4\)

\(=\)

\(\ds \map \beta 1\)

Definition of Dirichlet Beta Function

$\blacksquare$