Determine that if $a_n \longrightarrow L$ then $\sqrt[n]{a_1\cdot...a_n}\longrightarrow L$
135155
Why is $\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \cdots a_n}} =\lim_{n \to +\infty}{a_n}$
89325
$a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
80960
If $a_n>0$ converges to $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a.$
82591
Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
75639
Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
88425
Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
12301
Show that $\lim_{n \rightarrow \infty} \sqrt[n]{a} =1$
63124
Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
42292
Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
12031
Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
50041
Significance of $\sqrt[n]{a^n} $?!
105110
Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?
153763
Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
68609
Let $a_{n+1}=\sqrt{a_1+a_2+\cdots+a_n}$ .Prove that $ \lim\limits_{n \rightarrow \infty} \frac{a_n}{n}=\frac{1}{2}$
108429
Show that if $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
107774
Prove $\|x\|_1\le \sqrt n \|x\|_2$
78144
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?