Question Number 8277 by Yozzias last updated on 05/Oct/16
![Show that one representation for π≈3.14... is π=12cos^(−1) [((3/4))^(1/4) (1+Σ_(r=1) ^∞ ((Π_(k=1) ^(2r) ((3/2)−k))/((2r)!))(((−1)/3))^r )].](https://www.tinkutara.com/question/Q8277.png)
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{one}\:\mathrm{representation}\:\mathrm{for}\:\pi\approx\mathrm{3}.\mathrm{14}… \\ $$$$\mathrm{is}\:\pi=\mathrm{12cos}^{−\mathrm{1}} \left[\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{1}/\mathrm{4}} \left(\mathrm{1}+\underset{\mathrm{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2r}} {\prod}}\left(\frac{\mathrm{3}}{\mathrm{2}}−\mathrm{k}\right)}{\left(\mathrm{2r}\right)!}\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{r}} \right)\right]. \\ $$$$ \\ $$