Question:
Show with generating functions that every positive integer has a unique decimal representation.
Attempt:
I've come up with the following to represent the ones, tens, hundreds, so on:
$(1+x^1+x^2+...+x^9)(1+x^{10}+x^{20}+..+x^{90})(1+x^{100}+...+x^{900})$.
I then get this for my generating function:
$\frac{(1-x^{10})(1-x^{100})(1-x^{1000})...}{(1-x)(1-x^{10})(1-x^{100})...}$
Even then, I'm not sure if I'm doing the problem correctly.
Edit:
Just realized that the terms cancel out so I get:
$\frac{1}{(1-x)}$
I'm stuck here now. I know I need to get something like $12 = 1*10^1 + 2*10^0$, but I don't know how to do that with generating functions.