<s>[INST] Can $(1-\frac{1}{2})(1-\frac{1}{2^2})(1-\frac{1}{2^3})...(1-\frac{1}{2^{n-1}})(\frac{1}{2^n})$ be simplified? Can $(1-\frac{1}{2})(1-\frac{1}{2^2})(1-\frac{1}{2^3})...(1-\frac{1}{2^{n-1}})(\frac{1}{2^n})$ be simplified? It seems like an expression from a simple induction proof problem that's missing its result.
 [/INST] You could write
$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{7}{8} \cdot \frac{15}{16} \cdot \dots \cdot \frac{2^{n - 1} - 1}{2^{n - 1}} \cdot \frac{1}{2^n}$$
and then the denominator is
$$2 \cdot 4 \cdot \dots 2^n = 2^{1 + 2 + 3 + \dots + n} = 2^{n(n + 1)/2}$$
while the numerator is
$$(2 - 1) (2^2 - 1) (2^3 - 1) \cdots (2^{n - 1} - 1) = \prod_{k = 1}^n (2^k - 1) = (-1)^n (2; 2)_n$$
where $(2; 2)_n$ is a $q$-Pochhammer symbol. So the final "simplification" would be
$$\frac{(-1)^n (2; 2)_n}{2^{n(n + 1)/2}}$$
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