query: Proof for $(n!)^2 \geq n^n$ without using induction
positive_passages.docid: 145891
positive_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 67527
negative_passages.text: Proof by Induction that $3^n ≥ 1+2^n$
negative_passages.docid: 102738
negative_passages.text: Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction.
negative_passages.docid: 55273
negative_passages.text: Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$
negative_passages.docid: 44864
negative_passages.text: Prove, formally that: $\log_2 n! \ge n$ , for all integers $n>3$.
negative_passages.docid: 72735
negative_passages.text: Understanding the proof of a formula for $p^e\Vert n!$
negative_passages.docid: 27580
negative_passages.text: Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
negative_passages.docid: 9920
negative_passages.text: $(n^2 - 1)!$ is a multiple of $(n!)^n$
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 66387
negative_passages.text: Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
negative_passages.docid: 153123
negative_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 41784
negative_passages.text: Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 20271
negative_passages.text: Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
negative_passages.docid: 86479
negative_passages.text: How to Prove with Mathematical Induction $3^n > n^2$
negative_passages.docid: 88838
negative_passages.text: How can I prove this inequality for $n\geq 2$?
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$