- Viète's formula
:"This article is not about
Viète's formulas for symmetric polynomials."Inmathematics , the Viète formula, named afterFrançois Viète , is the followinginfinite product type representation of the mathematical constant π::
The above formula is now considered as a result of one of
Leonhard Euler 's formula - branded more than one century after.Euler discovered that::
Substituting x=π/2 will produce the formula for 2/π, that is represented in an elegant manner by Viète.
The expression on the right hand side has to be understood as a limit expression
:where with initial condition
(Wells 1986, p. 50; Beckmann 1989, p. 95). However, this expression was not rigorously proved to converge until Rudio (1892).
Upon simplification, a pretty formula for π is given by
:
(J. Munkhammar, pers. comm., April 27, 2000).
Proof
Using an iterated application of the double-angle formula
:
for
sine one first proves the identity :valid for all positive integers "n". Letting "x=y/2n" and dividing both sides by cos("y"/2) yields
:
Using the double-angle formula sin "y"=2sin("y"/2)cos("y"/2) again gives
:
Substituting "y"=π gives the identity
:
It remains to match the factors on the right-hand side of this identity with the terms "an". Using the half-angle formula for
cosine ,:
one derives that satisfies the recursion with initial condition . Thus "an=bn" for all positive integers "n".
The Viète formula now follows by taking the limit "n" → ∞. Note here that
:
as a consequence of the fact that (this follows from
l'Hôpital's rule ).π
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