# Infinite product

When 0 < x < 1, we can be sure that the following infinite product converges.

$$$\prod_{n=1}^\infty (1 + x^n)$$$

But is there a closed form for it?

Not really an answer to your question, but the expression you've post is the way of generating the partitions of integers.

Write out the first few terms, $$(1+x)(1+x^{2})(1+x^{3})(1+x^{4})(1+x^{5})(1+x^{6})(1+x^{7})$$

You can see that the coefficent of the constant term in the expansion will be just 1.
Coefficent of the x term will be 1
Coefficent of the x^2 term will be 1
Coefficent of the x^3 term will be 2, because x^3 can be made from $x.x^{2}$ or just x^3 (ie 3 = 3+0 or 2+1)
x^4 will have coefficent 2 because x^4 comes from x^4 and x^3 * x (ie 4 = 4+0 or 3+1)
x^5 is 3, because 5 = 5+0 or 5=4+1 or 5=3+2
x^6 is 4 because 6=6+0 or 6=5+1 or 6=4+2 or 6 = 3+2+1)

See how it works? The coefficent of x^n will be the number of ways you can express n as a sum of unique integers. The closed form for this I don't think exists.

Well, this is interesting. It appears that I am looking for the generating function of a very particular sequence, although I hadn't realize that. So finding a closed form seems against the odds...