Convergent Infinite Product in Proof of Partitions Identity

Question

I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998).

Theorem 1.1 is a standard result that writes the generating function of a partition as an infinite product, e.g.

$$ \sum_{n \geq 0} p(n) q^n = \prod_{n \geq 0} \frac{1}{1 - q^n}. $$

At the end of the proof there is a discussion of convergence on page 5 of my copy.

The claim is made that if $0 < q < 1$ then $$ \prod_{i = 1}^\infty \frac{1}{1 - q^i} < \infty $$

Andrews seems to think this statement is obvious, but I don't see the justification. Could someone please point me in the right direction?

Answer 1

The convergence criteria for the infinite product is:

$\prod_\limits{n=1}^{\infty} a_n$ converges to a nonzero real number if and only if the sum $\sum_\limits{n=1}^{\infty} \log(a_n)$ converges.

$\log(a_n)$ in this case is $\log(\frac{1}{1-q^i})=-\log(1-q^i)$.

This can be expanded because $|q^i|\lt1$, and becomes $q^i+\frac{q^{2i}}{2}+\frac{q^{3i}}{3}+\dots$.

Sum all these together, and collect terms by the fraction to give:

$\frac{q}{1-q}+\frac{q^2}{2(1-q^2)}+\frac{q^3}{3(1-q^3)}+\dots\lt \zeta(1)$

and therefore converges by the PNT.