Factorial estimate
Task number: 3847
Prove the following factorial estimate: \( n^{n/2} \le n! \le \left(\frac{n+1}{2}\right)^n \)
Solution
We use \( (n!)^ 2 = \prod\limits_{i = 1}^n i (n + 1-i) \).
The lower bound follows from \( i (n + 1-i) \ge n \), because for \( i = 1 \) or \( n \) equality applies and in other cases is less than at least 2 and greater than at least \( \frac{n}2 \).
The upper estimate follows from the AG inequality \( \sqrt{ab} \leq \frac12 (a + b) \), i.e. \( i (n + 1-i) \leq \left (\frac {n + 1} 2 \right)^2 \).
