Let:

A₀ = 0
A₁ = √(1/2 + 1/2*A₀)
A₂ = √(1/2 + 1/2*A₁)
A₃ = √(1/2 + 1/2*A₂)
...
A_n = √(1/2 + 1/2*A_n-1)
...

Evaluate, analytically, the infinite product

P = A₁ * A₂ * A₃ * ...

A(1)=sqrt(1/2)=Cos(pi/4)

A(n)=sqrt( (A(n-1)+1)/2 )

we also have Cos(x/2)=sqrt( (Cos(x)+1)/2 )

putting these togeather we get

A(n)=Cos(pi/( 2^(n+1)))

that makes P=Cos(pi/4)*Cos(pi/8)*.....

now eulers infinite product states that Cos(t/2)*Cos(t/4)*.....=Sin(t)/t

if we let t=pi/2 we get P and thus P=Sin(pi/2)/(pi/2)=2Sin(pi/2)/pi

thus P=2/Pi

Posted by Daniel on 2008-06-07 15:40:35