Prove $\sqrt n\leq \sqrt[n]{n!} \leq \frac{n+1}{2}$
132732
Estimate of n factorial: $n^{\frac{n}{2}} \le n! \le \left(\frac{n+1}{2}\right)^{n}$
26876
How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
80920
Prove or disprove: For any $v\in\mathbb{R}^n, \|v\|_1\|v\|_\infty \leq \frac{1+\sqrt{n}}{2}\|v\|_2^2$
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
87411
Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
52625
Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
128882
Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
45914
Proving that $n!≤((n+1)/2)^n$ by induction
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
61487
Limit of $\frac1n\sqrt[n]{(n+1)(n+2)(n+3)...(n+n)}$
96332
Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
49750
Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$
16798
Prove the inequality $n! \geq 2^n$ by induction
118244
An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$