Limit of $S_n=\sqrt[n(n+1)]{\prod _{k=0}^{n}\binom{n}{k}}$
43188
The $n^{th}$ root of the geometric mean of binomial coefficients.
10902
Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
81654
Evaluate $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]$ where $k<p$
84384
Evaluate $\lim_{n\to \infty}\left(\frac{1}{\sqrt{n(n+1)}}+\frac{1}{\sqrt{(n+1)(n+2)}}+\cdots+\frac{1}{\sqrt{(2n-1)2n}}\right)$
46103
Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$
104704
Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
131425
Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
96361
Limit of $n$th root of $n$!
60061
Find the value of : $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$
84595
How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?
60036
How to prove this limit exists: $a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_{n}}$
16970
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
7105
Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
148320
Find the limit $\lim_\limits{n\to{\infty}}n^2(\sqrt[n]{x}-\sqrt[{n + 1}]{x})$ and my inquiry about Stolz-Cesàro theorem
6558
Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$
119145
$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$
109576
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$