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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 * ...

  Submitted by pcbouhid    
Rating: 5.0000 (1 votes)
Solution: (Hide)
Using the hint given, and the equality cos(2a) = 2*cos2(a) - 1, we arrive at the product:

P = cos(pi/4) * cos(pi/8) * cos(pi/16) * ...

To evaluate this, we use cos(a) = sin(2a)/2sin(a).

The product of the first n terms of P simplify to:

Pn = sin(pi/2) / [2n * sin(pi/2n+1)].

The limit of Pn as n tends to infinity, after making 2n+1 = pi/x, leads us to P = 2/pi.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: Analytical solution ======== what if?pcbouhid2008-06-08 02:11:44
Analytical solutionDaniel2008-06-07 15:40:35
re: Identity of numeric value ------ a first hintpcbouhid2008-06-06 20:03:37
Some ThoughtsIdentity of numeric valuePaul2008-06-06 16:03:28
Some ThoughtsFor what it's worth (numerical value)Charlie2008-06-05 12:27:29
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