query: Prove that $\prod\limits_{n= 2}^{\infty} \frac{n^{3}+ 1}{n^{3}- 1}= \frac{3}{2}$, which is a beautiful value!
positive_passages.docid: 1166
positive_passages.text: How to compute $\prod\limits^{\infty}_{n=2} \frac{n^3-1}{n^3+1}$
negative_passages.docid: 22636
negative_passages.text: Find the sum of the series $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$
negative_passages.docid: 60098
negative_passages.text: How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?
negative_passages.docid: 60197
negative_passages.text: Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
negative_passages.docid: 110588
negative_passages.text: Showing $\lim_{n\rightarrow\infty}\sqrt[3]{n^3+n^2}-\sqrt[3]{n^3+1}\rightarrow\frac{1}{3}$
negative_passages.docid: 123248
negative_passages.text: How can I show $\lim_{n\to\infty}\prod_{j}^n\frac{\sin(jn^{-3/2}u)}{jn^{-3/2}u}=e^{-u^2/18}$?
negative_passages.docid: 37484
negative_passages.text: Is there an elementary argument to show that $\prod_{n=0}^\infty \frac{2n+1}{2n+3}=0$
negative_passages.docid: 144310
negative_passages.text: How would you prove that $\lim\limits_{n\to\infty}(\sqrt{4n^2+n}-2n)=\frac14$?
negative_passages.docid: 22773
negative_passages.text: What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$
negative_passages.docid: 27579
negative_passages.text: Evaluating the infinite product $ \prod\limits_{n=1}^{\infty} \cos(\frac{y}{2^n}) $
negative_passages.docid: 13796
negative_passages.text: What is the value of $\prod_{n=1}^\infty (1+\frac{1}{n^2})$?
negative_passages.docid: 17713
negative_passages.text: What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?
negative_passages.docid: 47706
negative_passages.text: Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $
negative_passages.docid: 44551
negative_passages.text: What is the result of $ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $ and why?
negative_passages.docid: 141404
negative_passages.text: Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$
negative_passages.docid: 5436
negative_passages.text: How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \ $?
negative_passages.docid: 78473
negative_passages.text: Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$