Viète's formula
Infinite product converging to 2/π / From Wikipedia, the free encyclopedia
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In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π:
It can also be represented as
The formula is named after François Viète, who published it in 1593.[1] As the first formula of European mathematics to represent an infinite process,[2] it can be given a rigorous meaning as a limit expression[3] and marks the beginning of mathematical analysis. It has linear convergence and can be used for calculations of π,[4] 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[5] 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.