Linked Questions

6
votes
1answer
1k views

How to evaluate this infinite product: $\prod\limits_{n=1}^{\infty }{\left( 1-\frac{1}{{{2}^{n}}} \right)}$ [duplicate]

How to evaluate this one $$\prod\limits_{n=1}^{\infty }{\left( 1-\frac{1}{{{2}^{n}}} \right)}$$
5
votes
3answers
1k views

Convergence of an infinite product $\prod_{k=1}^{\infty }(1-\frac1{2^k})$? [duplicate]

Problem: I want to prove that the infinite product $\prod_{k=1}^{\infty }(1-\frac{1}{2^{k}})$ does not converge to zero. It doesn't matter to find the value to which this product converges, but I am ...
5
votes
3answers
283 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$? [duplicate]

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
6
votes
1answer
360 views

Evaluate $\prod_{n=1}^\infty \frac{2^n-1}{2^n}$

Is there a closed form expression for this limit? $$\prod_{n=1}^\infty \frac{2^n-1}{2^n}$$ Wolfram Alpha says $0.2887880950866024212788997219292307800889\dots$ and the Inverse Symbol Calculator found ...