Leibniz's Formula for Pi/Proof by Dirichlet Beta Function
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Theorem
- $\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$
That is:
- $\ds \pi = 4 \sum_{k \mathop \ge 0} \paren {-1}^k \frac 1 {2 k + 1}$
Proof
Recall the Dirichlet beta function:
- $\ds \map \beta s = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}$
From Dirichlet Beta Function at Odd Positive Integers, we obtain:
- $\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$
Therefore, setting $n = 0$ above:
\(\ds \map \beta 1\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^1}\) | Definition of Dirichlet Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^0 \dfrac {E_0 \pi } {4 \times 0!}\) | Dirichlet Beta Function at Odd Positive Integers | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4\) | Definition of Euler Numbers and Factorial of Zero |
$\blacksquare$