I was trying to figure out the expansion of the product:

$$Q= (1+x)(1+x^2)(1+x^4)..(1+x^{2^n})$$

Multiplying and dividing by $1-x$

$$ Q = \frac{1}{1-x} [ 1-x^{2^{n+1}}]$$

Expanding the denominator as geometric series:

$$ Q = - [ x^{2^{n+1}} - 1 ] [ \sum\_{j=0}^{\infty} x^j] = - [ \sum\_{j=0}^{\infty} x^{2^{n+1}+j} - x^{j}]$$

Now,

$$ \sum\_{j=0}^{\infty} x^j = \sum\_{j=0}^{2^{n+1}} x^j + \sum\_{j=2^{n+1}}^{\infty} x^{j} = \sum\_{j=0}^{2^{n+1}} x^j + \sum\_{j=0}^{\infty} x^{2^n+j}$$

Using the above on $Q$,

$$ Q = \sum\_{j=0}^{2^{n+1}} x^j$$

Or,

$$ (1+x)(1+x^2)(1+x^4)..(1+x^{2^n})=\sum\_{j=0}^{2^{n+1}} x^j$$

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My question: $ \frac{1}{1-x} = \sum\_{j=0}^{\infty} x^j$ only holds if $ |x|<1$ so this chain of logical implications? But it is weird to me because I had a finite polynomial end ended in a finite polynomial so I am pretty convinced that my final polynomial is actually the expansion of first.. so, how do I justify this gives me the equation which holds for all $ x \in R$