Let $P_n$ be a polygon inscribed in a circle with diameter $1$. Each side of the polygon has length $l_n=\sin(\pi /n)$ and the circumference of $P_n=nl_n$. With $$P_n= \pi - \frac{\pi^3}{3!}\frac{1}{n^2}+ \frac{\pi^5}{5!}\frac{1}{n^4}- \frac{\pi^7}{7!}\frac{1}{n^6}+\dots$$ We have $P_2=2,~P_3=\frac{3\sqrt{3}}{2},~P_4=2\sqrt{2},P_6=3$.
Extrapolate using the values $P_2$,...,$P_4$ and give estimates and the error for $\pi$.
$$P_2= \pi - \frac{\pi^3}{3!}\frac{1}{2^2}+ \frac{\pi^5}{5!}\frac{1}{2^4}- \frac{\pi^7}{7!}\frac{1}{2^6}+O(h^8)$$ $$P_3= \pi - \frac{\pi^3}{3!}\frac{1}{3^2}+ \frac{\pi^5}{5!}\frac{1}{3^4}- \frac{\pi^7}{7!}\frac{1}{3^6}+O(h^8)$$ $$P_4= \pi - \frac{\pi^3}{3!}\frac{1}{4^2}+ \frac{\pi^5}{5!}\frac{1}{4^4}- \frac{\pi^7}{7!}\frac{1}{4^6}+O(h^8)$$ $$P_6= \pi - \frac{\pi^3}{3!}\frac{1}{6^2}+ \frac{\pi^5}{5!}\frac{1}{6^4}- \frac{\pi^7}{7!}\frac{1}{6^6}+O(h^8)$$
To remove the first error term I extrapolarted $\frac{4P_4-P_2}{3}$ and $\frac{4P_6-P_3}{3}$, and got approximations with errors $1.17838\%$ and $0.00762\%$ respectively. How do I extrapolate $P_2$ with $P_3$ and further to eliminate the 2nd and 3rd order terms?
