Originally Posted by

**danny arrigo** More details on the last one and it's answer. Consider

$\displaystyle (1-x^4)(1+x^4) = 1 - x^8$

$\displaystyle \underbrace{(1-x^4)(1+x^4)}_{\text{from above} \, = \,(1-x^8)}(1+x^8) = (1-x^8)(1+x^8) = 1 - x^{16}$

$\displaystyle \underbrace{(1-x^4)(1+x^4)(1+x^8)}_{\text{from above} \, = \,(1-x^{16})}(1+x^{16}) = (1-x^{16})(1 + x^{16}) = 1- x^{32}$

which generalizes (and can be proven by induction)

$\displaystyle (1-x^4)(1+x^4)(1+x^8) \cdots (1+x^{2N}) = 1 - x^{2(N+1)}$.

Hence,

$\displaystyle (1+x^4)(1+x^8) \cdots (1+x^{2N}) = \frac{1- x^{2(N+1)}}{1-x^4}$,

or

$\displaystyle \prod_{n=2}^N 1+x^{2n} = \frac{1- x^{2(N+1)}}{1-x^4}$.

If $\displaystyle |\,x\,| <1$, then in the limit as $\displaystyle N\, \rightarrow\, \infty$

$\displaystyle \prod_{n=2}^{\infty} 1+x^{2n} = \frac{1}{1-x^4}$.

(Sorry, I just couldn't leave it without explanation)