Viète's formula

In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant $π$ :

{\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots

It can also be represented as:

{\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}

The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a limit expression, and marks the beginning of mathematical analysis. It has linear convergence, and can be used for calculations of $π$ , but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses, and as a motivating example for the concept of statistical independence.

The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.

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