A question about an infinite product
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Oct16-11, 10:42 AM
I am puzzled by the following infinite product:
Let B > A
A - B = [A^(1/2) + B^(1/2)] * [A^(1/2) - B^(1/2)]
= [A^(1/2) + B^(1/2)] * [A^(1/4) + B^(1/4)] * [A^(1/4) - B^(1/4)]
=[A^(1/2) + B^(1/2)] * [A^(1/4) + B^(1/4)] * [A^(1/8) + B^(1/8)] * [A^(1/8) - B^(1/8)]
Continuing the obvious expansion into an infinite product produces a sequence of terms none of which are negative. However, since B > A then A - B is a negative value. How can a infinite product of terms greater than zero produce a negative value? Surely there is a "simple" explanation.
Thanks for any comments.
In each of the products the last term to the right is negative.