<s>[INST] A question about inequality ${(n+1)\over e^n}^nHow to prove the inequality $${(n+1)\over e^n}^n<n!$$
I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
 [/INST] Use the binomial theorem on $(1 + n)^n$ and try.
$$(1 + n)^n = 1 + \frac{n!}{1!(n - 1)!} n + \frac{n!}{2!(n - 2)!} n^2 + \frac{n!}{3!(n - 3)!} n^3 + \dots + \frac{n!}{n!(n - n)!} n^n \\ = n! \{\frac{1}{0! n!} + \frac{n}{1! (n-1)!} + \frac{n^2}{2! (n-2)!} + \frac{n^3}{3! (n-3)!} + \dots + \frac{n^n}{n! (n-n)!}\} \\ \le n! \{1 + \frac{n}{1!} + \frac{n^2}{2!} + \frac{n^3}{3!} + \dots\} \\ = n! e^n$$
Hope it is clear now.
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