I came across a problem that asked me to prove this for $|x|<1$. My first instinct upon seeing the identity was the fact that $\frac{1}{1-x}$ happens to be the generating function of the sequence $1, 1, 1, \dots$, corresponding to the formal series $1+x+x^2+\dots$. Although I proved the identity without resorting to any combinatorics (using a telescoping product, as discussed here), it would be interesting to know if
$$ \prod^\infty_{k=0}\left( 1+x^{2^k}\right) = \frac{1}{1-x} = \sum^\infty_{k=0}x^k$$ has a combinatorial interpretation related to generating functions.
Best Answer
If you want to find the coefficient of $x^n$ in the expression $$(1+x)(1+x^2)(1+x^4) (1+x^8)\dots,$$ you could reframe the question as "How many ways are there to write $n$ as the sum of powers of $2$, using each power only once?"
This is just asking about how many ways there are to express $n$ as a binary number, which is $1$.