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词汇 Leibniz Series
分类 英语词汇 英语翻译词典
释义

Leibniz Series

英语百科

Leibniz formula for π

Comparison of the convergence of the Leibniz formula (cyan squares) and several historical infinite series for π. Sn is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)
See List of things named after Gottfried Leibniz for other formulas known under the same name.

In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that

1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}.\!

Using summation notation:

\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} \;=\; \frac{\pi}{4}.\!
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