query: What is the value of $\prod_{i=1}^\infty \left(1-\frac{1}{2^i}\right)$?
positive_passages.docid: 24695
positive_passages.text: How to solve the limit $\lim\limits_{n\to \infty}(1-a)(1-a^2)\cdots(1-a^n)$?
negative_passages.docid: 141404
negative_passages.text: Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$
negative_passages.docid: 7105
negative_passages.text: Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
negative_passages.docid: 72904
negative_passages.text: Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$
negative_passages.docid: 138683
negative_passages.text: Proving $\prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right) = \frac{n+1}{2n}$ by induction
negative_passages.docid: 115715
negative_passages.text: Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$
negative_passages.docid: 158357
negative_passages.text: Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
negative_passages.docid: 105697
negative_passages.text: How do I evaluate $\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?
negative_passages.docid: 60061
negative_passages.text: Find the value of : $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$
negative_passages.docid: 108010
negative_passages.text: Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$
negative_passages.docid: 27579
negative_passages.text: Evaluating the infinite product $ \prod\limits_{n=1}^{\infty} \cos(\frac{y}{2^n}) $
negative_passages.docid: 78008
negative_passages.text: A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$
negative_passages.docid: 43509
negative_passages.text: Evaluating $\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$ for $x\in\mathbb{R}^{+}$
negative_passages.docid: 146221
negative_passages.text: Find the product $\prod_{n=1}^{\infty} (1+x^{2n-1}) $
negative_passages.docid: 18285
negative_passages.text: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
negative_passages.docid: 3527
negative_passages.text: Finding $\lim_{n \to \infty} \prod_{r=1}^n (1+ \frac{1}{a_r})$
negative_passages.docid: 117911
negative_passages.text: What's the limit of $\prod_1^\infty \left(1-\frac{1}{2^n}\right)=(1-1/2)(1-1/4)(1-1/8)...$?