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.

It can be transformed into
$$
\frac{(2-1)(2^2-1)(2^3-1)\cdots(2^{n-1}-1)}{2^{1+2+3+\cdots +n}}=
\frac{(2-1)(2^2-1)(2^3-1)\cdots(2^{n-1}-1)}{2^{n(n+1)/2}}
$$
but whether this is a simplification I don't know.
