How to prove $n!\leq(\frac{n+1}{2})^n$
35611
How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
25458
Prove inequalities $2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + .... + 1/\sqrt{n}\leq 2\sqrt{n}-1$
150642
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
54369
Proof of $n^2 \leq 2^n$.
2554
Show that if $n>2$, then $(n!)^2>n^n$.
136403
Prove that $\sqrt{a}\leq\frac{1+a}{2}$
7105
Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
80216
Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
26962
Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$
71014
Prove that 2(nC0+nC2...nC$\frac{(N-1)}{2}$=$2^n$
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
69921
Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
102334
How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
80556
Demonstrate by induction the inequality: $\ln(1+n)\leq\sum_{i=1}^{n}\frac{1}{i}\leq1+\ln(n)$
10629
How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$