query: limit of product of $(a_1a_2.\dots a_n)^{\frac{1}{n}}$
positive_passages.docid: 14013
positive_passages.text: How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)... 2n} = \frac{4}{e}$
negative_passages.docid: 90496
negative_passages.text: Proving $\lim \frac{x^n-a^n}{x-a}=n\cdot a^{n-1}$
negative_passages.docid: 83318
negative_passages.text: How to compute the limit $\lim_{n \to \infty} \sqrt[n]{a_1^n + \cdots + a_m^n}$?
negative_passages.docid: 17090
negative_passages.text: Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$
negative_passages.docid: 26300
negative_passages.text: $a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $ \lim_{n\to \infty}\frac{s_1+s_2+\dots s_n}{n}$
negative_passages.docid: 97047
negative_passages.text: Limit for sequence $a_{m+n}\leq a_m+a_n$
negative_passages.docid: 3527
negative_passages.text: Finding $\lim_{n \to \infty} \prod_{r=1}^n (1+ \frac{1}{a_r})$
negative_passages.docid: 10902
negative_passages.text: Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
negative_passages.docid: 18152
negative_passages.text: Find the limit points of the set $\{ \frac{1}{n} +\frac{1}{m} \mid n , m = 1,2,3,\dots \}$
negative_passages.docid: 155353
negative_passages.text: Find the limit of $ a_n =\left(1+\frac{1^2}{n^2}\right)^1\left(1+\frac{2^2}{n^2}\right)^2\cdots \left(1+\frac{n^2}{n^2}\right)^n$
negative_passages.docid: 83280
negative_passages.text: Infinite product $ \prod^{\infty}_{n=0}\left(1-\frac{1}{2^{3n}}-\frac{1}{2^{6n}}\right)$
negative_passages.docid: 107764
negative_passages.text: Limit series $(a^n-b^n)^{1/n}$
negative_passages.docid: 31174
negative_passages.text: Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
negative_passages.docid: 32853
negative_passages.text: Computing $\lim_{A\to\infty} \frac{1}{A} \int\limits_1^A \! A^{\frac{1}{x}} \, \mathrm{d}x.$
negative_passages.docid: 77605
negative_passages.text: Limit of $\prod\limits_{k=2}^n\frac{k^3-1}{k^3+1}$
negative_passages.docid: 96366
negative_passages.text: show that for any n $\in$ $\Bbb N$ , if n $\ge$ 2, then $\prod_\limits{i=2}^n $ $\left( 1-\frac{1}{i^2}\right)$ = $\frac{n+1}{2n}$
negative_passages.docid: 97720
negative_passages.text: Asymptotic behavior of $\sum\limits_{k=1}^n \frac{1}{k^{\alpha}}$ for $\alpha > \frac{1}{2}$