Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1/Proof 2
Jump to navigation
Jump to search
Theorem
- $\map \beta 1 = \dfrac \pi 4 $
Proof
\(\ds \frac 1 {1 + x^2}\) | \(=\) | \(\ds 1 - x^2 + x^4 - x^6 + \cdots\) | 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 |
$\blacksquare$