lean_workbook_plus_37686

In the following let \begin{array}{|c|}\hline A=2\ \hline \end{array}\; and $P_0=q_0=\frac{3}{2}\; .$ \ni) Using a PC , find $p_k$ and $q_k$ for $k\in \{1,2,3,4,5\}.$ \nii) Show that \n\n $\begin{array}{lcl} \displaystyle p_n&=&\displaystyle F_{{}_{2^n}}(\beta)=\displaystyle \sqrt{2}\frac{1+\beta^{2^n}}{1-\beta^{2^n}}\ &&\ \displaystyle q_n&=&\displaystyle F_{{}_{5^n}}(\beta)=\displaystyle \sqrt{2}\frac{1+\beta^{5^n}}{1-\beta^{5^n}} \end{array}\; \; , \; \; \; \beta: =\frac{3-2\sqrt{2}}{3+2\sqrt{2}}=\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)^2< \frac{1}{5^2}\; .$ \niii) Prove the inequalities \n $\begin{array}{lcl} 0<p_5-\sqrt{2}&<& \displaystyle \frac{1}{{10}^{42}} \ &&\ 0<q_5-\sqrt{2}&<& \displaystyle \frac{1}{{10}^{10416}} \ \end{array}\; .$ \nThis means that $p_5 \approx \sqrt{2}$ , both numbers having in common at least 41 decimals. On the other hand, $q_5$ approximate $\sqrt{2}$ with at least 10415 exact decimals ! \nNote that $\sqrt{2}=1.4142\; 13562\; 37309\; 50488...$