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Evaluate this infinite product (Posted on 2008-06-05) Difficulty: 3 of 5
Let:

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

Evaluate, analytically, the infinite product

P = A1 * A2 * A3 * ...

See The Solution Submitted by pcbouhid    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Analytical solution | Comment 4 of 5 |

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
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