Your first two can be found here Infinite Product -- from Wolfram MathWorld
and your third can be written in closed form
Hi, I've been asked to find the value of some infinite products. We've done infinite series before, but not products. There's a few questions - if someone could help me out with the first one, I could try the other two myself.
Question: Find the values of the following infinite products:
Cheers!
Your first two can be found here Infinite Product -- from Wolfram MathWorld
and your third can be written in closed form
You know how you can sometimes sum a series by forming a "telescoping sum", where you manipulate the terms (maybe by using partial fractions) so that each term cancels with those on either side of it? Well, to evaluate an infinite product, you can sometimes form a "telescoping product", where something similar happens.
For example, . Therefore
as .
For you can do the same thing. Factorise as , write down the product of the first few terms and see how they telescope. You should then be able to show that the product of the first N terms converges to 2/3 as N→∞.