Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1/Proof 2

Theorem

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

$\ds \frac 1 {1 + x^2}$

$=$

$\ds 1 - x^2 + x^4 - x^6 + \cdots$

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

$=$

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

integrating both sides from $0$ to $1$

$\ds \leadsto \ \ $

$\ds \map \arctan 1 - \map \arctan 0$

$=$

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

$\ds \leadsto \ \ $

$\ds \frac \pi 4 - 0$

$=$

$\ds 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots$

$\ds \leadsto \ \ $

$\ds \frac \pi 4$

$=$

$\ds \map \beta 1$

$\blacksquare$