How to compute the infinite product? Given the following infinite product:
$$\lim_{n\to \infty} \frac{1}{2}\frac{3}{4}\cdots \frac{2^n-1}{2^n}.$$
It is easy to see that above infinite product is convergent by the convergence of the following series:
$$\sum_{n=1}^{\infty}\ln \left(1-\frac{1}{2^n}\right)$$.
But I do not know how to calculate the infinite product. I would appreciate it if someone can give any suggestions and comments.

You maybe could simplify the denominator of the infinite product since $\prod_{n=1}^{m}2^n=2^{\sum_{n=1}^{m}n}=2^{\frac{m(m+1)}{2}}$ which would mean that $\lim_{m \rightarrow \infty} \prod_{n=1}^{m}\frac{2^n-1}{2^n}=\lim_{m \rightarrow \infty}\frac{1}{2^{\frac{m(m+1)}{2}}} \prod_{n=1}^{m}(2^n-1)$
