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Limit of a particular variety of infinite product/series

Define $$F(x) = \prod_{n=1}^\infty(1-x^n)$$ where $|x|<1$.

How can one compute $F(1/2)$? (Without an obvious polynomial expansion or brute-force calculation.)

This is sometimes called Euler's function.

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marked as duplicate by Raymond Manzoni, userNaN, Henry T. Horton, TMM, Austin Mohr Dec 13 '12 at 19:54

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Some related information (no 'closed form'). –  Raymond Manzoni Dec 13 '12 at 18:02
Quick simulation stops around 0.288788... –  gt6989b Dec 13 '12 at 18:07
@RaymondManzoni that's a great source. Nearly as good as a closed form. –  santa claus Dec 13 '12 at 18:10
Glad it helped @AlecS but I fear we will have to close this as duplicate! –  Raymond Manzoni Dec 13 '12 at 18:16

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up vote 2 down vote accepted

That product is not so simple as you think. Euler proved that $$F(x) = \prod_{n=1}^\infty(1-x^n)=\sum_{-\infty}^{+\infty}(-1)^kx^{\frac{k(3k+1)}{2}}$$ This problem arises in partition number theory and is called Euler's pentagonal number theorem.

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Thanks a lot. I hadn't seen this identity. –  santa claus Dec 13 '12 at 18:25
@AlecS: This paper from Bell may help you too. –  Raymond Manzoni Dec 13 '12 at 18:29
@RaymondManzoni I had begun reading it already. :) –  santa claus Dec 13 '12 at 18:36
@AlecS: a rather fascinating subject I'll admit... –  Raymond Manzoni Dec 13 '12 at 18:45

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