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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.


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

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

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.

  • $\begingroup$ @AlecS: This paper from Bell may help you too. $\endgroup$ – Raymond Manzoni Dec 13 '12 at 18:29
  • $\begingroup$ @RaymondManzoni I had begun reading it already. :) $\endgroup$ – Pricklebush Tickletush Dec 13 '12 at 18:36
  • $\begingroup$ @AlecS: a rather fascinating subject I'll admit... $\endgroup$ – Raymond Manzoni Dec 13 '12 at 18:45

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