Find the infinite product

First, we convert all of these to powers of two: , so that our product is . And by the basic law of exponents , we thus have

Now, one way to find this infinite sum is to convert it into an infinite sum of infinite geometric series:

And using the infinite geometric series formula with our , we see

Which is itself a geometric series with sum

And thus our product is

There is also another way we could have found our sum, using calculus. Noting that

for , we see that

for

Now, taking the derivative of both sides with respect to *x*:

But the left hand sum becomes our desired sum when *x*=1, so making that substitution, we see ,

the same result we found above.

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