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Question Number 8277 by Yozzias last updated on 05/Oct/16

$$\mathrm{Show}\mathrm{that}\mathrm{one}\mathrm{representation}\mathrm{for}\pi \approx 3.14...$$$$\mathrm{is}\pi ={12\cos }^{-1}\left[{\left(\frac{3}{4}\right)}^{1/4}(1+\underset{\mathrm{r}=1}{\sum ^{\mathrm{\infty }}}\frac{\underset{\mathrm{k}=1}{\prod ^{2\mathrm{r}}}(\frac{3}{2}-\mathrm{k})}{\left(2\mathrm{r}\right)!}{\left(\frac{-1}{3}\right)}^{\mathrm{r}})\right].$$$$$$

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