Chapter 11.5, Problem 4E

To determine

Find approximations for $\pi$ .

Explanation of Solution

Given:

  $\pi \approx 2÷\left(\sqrt{0.5}\cdot \sqrt{0.5+0.5\sqrt{0.5}}\cdot \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}\cdot \mathrm{...}\right)$

Calculation:

Calculate

  $\begin{array}{l}\sqrt{0.5}=0.707\\ 0.5\sqrt{0.5}=0.35355\\ \sqrt{0.5+0.5\sqrt{0.5}}=0.9238795\\ 0.5\sqrt{0.5+0.5\sqrt{0.5}}=0.4619397\\ \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}=0.9807852\\ 0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}=0.99039264\\ \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}}=0.995184726\\ \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}}}=0.998795456\end{array}$

Now,

  $\begin{array}{l}\sqrt{0.5}\cdot \sqrt{0.5+0.5\sqrt{0.5}}\cdot \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}\cdot \\ \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}}\cdot \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}}}\\ =(0.707)(0.9238795)(0.9807852)(0.995184726)(0.998795456)\\ =0.636779257\end{array}$

  $\begin{array}{l}2÷\left(\sqrt{0.5}\cdot \sqrt{0.5+0.5\sqrt{0.5}}\cdot \sqrt{0.5+0.5\sqrt{0.5+0.5\sqrt{0.5}}}\cdot \mathrm{...}\right)\\ \approx \frac{2}{0.636779257}\approx 3.14\approx \pi \end{array}$

Hence proved.