query: prove that $(\frac{n}{3})^n<n!<e\cdot(\frac{n}{2})^n$
positive_passages.docid: 83149
positive_passages.text: Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$
negative_passages.docid: 44705
negative_passages.text: Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
negative_passages.docid: 90496
negative_passages.text: Proving $\lim \frac{x^n-a^n}{x-a}=n\cdot a^{n-1}$
negative_passages.docid: 7404
negative_passages.text: Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 5927
negative_passages.text: How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
negative_passages.docid: 155005
negative_passages.text: How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$
negative_passages.docid: 61963
negative_passages.text: Proving that $\sum\limits_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$
negative_passages.docid: 79847
negative_passages.text: proving of inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$
negative_passages.docid: 89265
negative_passages.text: Proving that $x - \frac{x^3}{3!} 0$
negative_passages.docid: 155984
negative_passages.text: Prove $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots$ converges to $\frac 1 2 $
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 725
negative_passages.text: proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
negative_passages.docid: 30495
negative_passages.text: Analysis: Prove divergence of sequence $(n!)^{\frac2n}$
negative_passages.docid: 67650
negative_passages.text: Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 62576
negative_passages.text: How to find $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$?