I am reading Keller & Trotter: Applied Combinatorics, pg. 155, and I am having trouble with an intermediate step in a proof.
The proof deals with integer partitions:
And the part I can't figure out is when the the author says:$$\prod_{n=1}^{\infty}(1 - x^n) = \prod_{n=1}^{\infty}(1 - x^{2n-1})\prod_{n=1}^{\infty}(1 - x^{2n})$$ ...at line two of the math content, next to the second equals sign aligned to the left:
How does he get there?
They’re simply splitting the product into factors with an odd exponent and factors with an even exponent:
$$\prod_{n\ge 1}\left(1-x^n\right)=\prod_{n\ge 1}\left(\left(1-x^{2n-1}\right)\left(1-x^{2n}\right)\right)=\prod_{n\ge 1}\left(1-x^{2n-1}\right)\prod_{n\ge 1}\left(1-x^{2n}\right)\;.$$
When $n=1$, for instance, $$\left(1-x^{2n-1}\right)\left(1-x^{2n}\right)=(1-x)\left(1-x^2\right)\;,$$ and when $n=2$, $$\left(1-x^{2n-1}\right)\left(1-x^{2n}\right)=\left(1-x^3\right)\left(1-x^4\right)\;.$$