# How can one compute this simple infinite product? [duplicate]

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.

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

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.