=+15. Prove the elementary inequalities ln n! − ln n 2 ≤

 n 1

ln t dt = n ln n − n + 1 ≤ ln n!

that point the way to Stirling’s formula. (Hint: Using the concavity of ln t, verify the inequality ln(m − 1) + ln m 2 ≤

 m m−1 ln t dt . )