I was just wondering if the following is legitimate or if there's a problem with it:
Using difference of squares,
$a^n - b^n = (a^{n/2}-b^{n/2})(a^{n/2}+b^{n/2}) = (a^{n/4}-b^{n/4})(a^{n/4}+b^{n/4})(a^{n/2}+b^{n/2}) $
Continuing expanding in the same way, in general, for any $k \in \mathbb{N}$, $a^n - b^n = (a^{n/{2^k}}-b^{n/{2^k}})(a^{n/{2^k}}+b^{n/{2^k}})(a^{n/{2^{k-1}}}+b^{n/{2^{k-1}}})\cdots(a^{n/2}+b^{n/2})$
So is it valid to say: $a^n - b^n$ = $\lim\limits_{x \to \infty} [(a^{n/{2^x}}-b^{n/{2^x}}) \cdot\prod_{k=1}^x (a^{n/{2^k}}+b^{n/{2^k}})]$ ?
If not, why not?
If so, the limit seems to evaluate to the indeterminate form $0\cdot\infty$ (I'm not so sure about this, but Wolfram Alpha says the infinite product on the right hand side of the $\cdot$ diverges), which makes me want to use L'Hôpital's rule, but I'm not so sure how to go about that. This might have no practical use, I was just playing around with this, but I just want to know if what I'm doing actually works mathematically.