F. Vieta(1540-1603) π = 2 × 2 2 × 2 2 + 2 × 2 2 + 2 + 2 × ⋯ \pi=2\times\frac{2}{\sqrt{2}}\times\frac{2}{\sqrt{2+\sqrt{2}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\times\cdots π=2×2 ​2​×2+2 ​ ​2​×2+2+2 ​ ​ ​2​×⋯

π 2 = 1 + 1 3 + 1 3 × 2 5 + 1 3 × 2 5 × 3 7 + 1 3 × 2 5 × 3 7 × 4 9 + ⋯ \frac{\pi}{2}=1+\frac{1}{3}+\frac{1}{3}\times\frac{2}{5}+\frac{1}{3}\times\frac{2}{5}\times\frac{3}{7}+\frac{1}{3}\times\frac{2}{5}\times\frac{3}{7}\times\frac{4}{9}+\cdots 2π​=1+31​+31​×52​+31​×52​×73​+31​×52​×73​×94​+⋯

J. Wallis(1616-1703) π 2 = 2 1 × 2 3 × 4 3 × 4 5 × 6 5 × 6 7 × ⋯ \frac{\pi}{2}=\frac{2}{1}\times\frac{2}{3}\times\frac{4}{3}\times\frac{4}{5}\times\frac{6}{5}\times\frac{6}{7}\times\cdots 2π​=12​×32​×34​×54​×56​×76​×⋯

J. Gregory(1638-1675)和莱布尼茨G. W. Leibniz(1646-1716) π 4 = 1 1 − 1 3 + 1 5 − 1 7 + 1 9 ⋯ \frac{\pi}{4}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\cdots 4π​=11​−31​+51​−71​+91​⋯ 即 π 4 = ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 \frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1} 4π​=n=0∑∞​2n+1(−1)n​ 收敛得很慢

马青公式John Machin(1686 –1751)于1706年发现