query: Show that $\left(2n\right)!\leq \left(n\left(n+1\right)\right)^{n}$.
positive_passages.docid: 145519
positive_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 57670
negative_passages.text: Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 4327
negative_passages.text: proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem
negative_passages.docid: 98768
negative_passages.text: Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
negative_passages.docid: 149056
negative_passages.text: Prove that for all $n\geqslant 1$ we have $F_n<{\left(\frac 74\right)}^n$.
negative_passages.docid: 41445
negative_passages.text: Show that the sequence $a_n\leq a_{2n}+a_{2n+1}$ diverges
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 2708
negative_passages.text: Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
negative_passages.docid: 99821
negative_passages.text: Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.
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: 71547
negative_passages.text: Prove $ 1+\left( 1-\frac{x}{n}\right)+...+\left(1-\frac{x}{n}\right)^{n-1}=\frac{n}{x}\left(1-\left(1-\frac{x}{n}\right)^n\right)$
negative_passages.docid: 39073
negative_passages.text: Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
negative_passages.docid: 16798
negative_passages.text: Prove the inequality $n! \geq 2^n$ by induction
negative_passages.docid: 70926
negative_passages.text: Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 104807
negative_passages.text: Prove the inequality $\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$