# Eikonal Blog

## 2010.01.05

### Pi calculations

Filed under: mathematics — Tags: , , , — sandokan65 @ 12:00

Bailey-Borwein-Plouffe formula:
$\pi = \sum_{n=0}^\infty \frac1{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac1{8n+5} - \frac1{8n+6} \right)$. (Sources: [2], [3]).

Plouffe:
$\frac{\pi}{8} = \sum_{n=0}^\infty \frac1{n 2^{\left[\frac{n+1}2\right]}} \left( \left[\frac{n+7}8\right] - \left[\frac{n+6}8\right] + \left[\frac{n+1}8\right] - \left[\frac{n+4}8\right]\right)$. (Source: [3])

Belard:
$\pi = \frac1{2^6} \sum_{n=0}^\infty \frac{(-)^n}{2^{10n}} \left( -\frac{2^5}{4n+1} - \frac1{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac1{10n+9} \right)$. (Source: [1])

Belard:
$\pi = \frac1{740,025}\left(\sum_{n=1}^\infty \frac{3P(n)}{\binom{7n}{2n} 2^{n-1}} - 20,379,280 \right)$ where:
$P(n) = - 885,673,181 n^5 + 3,125,347,237 n^4 - 2,942,969,225 n^3 + 1,031,962,795 n^2 - 196,882,274 n + 10,996,648$. (Source: [1])

Chudnovsky:
$\frac1\pi = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty (-)^k \frac{(6k)!}{(k!)^3 (3k)!} \frac{(13591409+545140134 k)}{640320^{3k}}$