Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Possible Duplicate:
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.

share|improve this question

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
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

1 Answer 1

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.

share|improve this answer
    
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

Not the answer you're looking for? Browse other questions tagged or ask your own question.