常數的近似（一）：圓周率

本文簡介如何以反正切函數計算圓周率的近似值。

1　圓周率的近似

1.1　簡易近似公式

設 $$\begin{array}{r}\alpha =\arctan \frac{1}{2}\text{與}\beta =\arctan \frac{1}{3}\text{。}\end{array}$$ 由 $0<\alpha +\beta <\pi /2$ 與 $$\begin{array}{rl}\tan (\alpha +\beta )& =\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\\ & =\frac{1/2+1/3}{1-1/6}\\ & =1\end{array}$$ 可知 $\alpha +\beta =\pi /4\text{。}$因此 $$\begin{array}{rl}\pi & =4(\arctan \frac{1}{2}+\arctan \frac{1}{3})\\ & =\sum _{n\ge 0}\frac{4(-1{)}^n}{2n+1}(\frac{1}{2^{2n+1}}+\frac{1}{3^{2n+1}})\text{。}\end{array}$$

若設 $$\begin{array}{rl}{a}_n& =\frac{4}{2n+1}(\frac{1}{2^{2n+1}}+\frac{1}{3^{2n+1}})\text{，}\end{array}$$ 並設 $$\begin{array}{rl}{S}_n& =\sum _{k=0}^n(-1{)}^k{a}_k\text{，}\end{array}$$ 則對任意非負整數 $n$ 均有 $S_{2n+1}<\pi <S_{2n}\text{。}$

1.2　計算近似值

我們可以用上述公式求出圓周率的近似值。首先我們枚舉 $((-1{)}^n{a}_n{)}_{n\ge 0}$ 中的一些項，有 $$\begin{array}{r}\begin{array}{rlrl}{a}_0& =& \frac{4}{1}(\frac{1}{2^1}+\frac{1}{3^1})=& \frac{10}{3}\text{，}\\ -a_1& =& \frac{-4}{3}(\frac{1}{2^3}+\frac{1}{3^3})=& \frac{-35}{162}\text{，}\\ a_2& =& \frac{4}{5}(\frac{1}{2^5}+\frac{1}{3^5})=& \frac{55}{1944}\text{，}\\ -a_3& =& \frac{-4}{7}(\frac{1}{2^7}+\frac{1}{3^7})=& \frac{-2315}{489888}\text{，}\\ a_4& =& \frac{4}{9}(\frac{1}{2^9}+\frac{1}{3^9})=& \frac{20195}{22674816}\text{，}\\ -a_5& =& \frac{-4}{11}(\frac{1}{2^{11}}+\frac{1}{3^{11}})=& \frac{-179195}{997691904}\text{。}\end{array}\end{array}$$ 接著依序加總可得 $$\begin{array}{r}\begin{array}{rlrl}{S}_0& =& a_0=& \frac{10}{3}\text{，}\\ S_1& =& S_0-a_1=& \frac{505}{162}\text{，}\\ S_2& =& S_1+a_2=& \frac{6115}{1944}\text{，}\\ S_3& =& S_2-a_3=& \frac{1538665}{489888}\text{，}\\ S_4& =& S_3+a_4=& \frac{498668825}{158723712}\text{，}\\ S_5& =& S_4-a_5=& \frac{21940173935}{6983843328}\text{。}\end{array}\end{array}$$ 此時由 $$\begin{array}{r}\frac{6283}{2000}<S_5<\pi <S_4<\frac{1257}{400}\end{array}$$ 可知 $3.1415<\pi <3.1425\text{，}$故圓周率 $\pi$（四捨五入至小數點後第三位）的近似值為 3.142。