Viète formula for ∏
François Viète, privy councillor to Henry III and IV of France, published this formula in 1593:
![\[\frac{2}{\pi}=\lim_{n\to\infty}\prod_{i=1}^{n}\frac{a_i}{2}\]](https://oopstart.com/wp-content/ql-cache/quicklatex.com-41f87f3bcb4c8871f0210509d1c48754_l3.png "Rendered by QuickLaTeX.com")
where
which unfolds as
![\[\frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}\cdot...\]](https://oopstart.com/wp-content/ql-cache/quicklatex.com-8bb1add037659399f10527a9b3ce53d8_l3.png "Rendered by QuickLaTeX.com")
This formula is very unwieldy for computation because:
- It is product rather than a sum
- Each term contains an increasing number of sub-terms
- Those sub-terms are square roots
- Those square roots are nested, meaning they cannot be parallelised
