Problem 340 (difficulty: 7/10)

Find positive constants \(\displaystyle c_1,c_2\) for which

\(\displaystyle c_1 \cdot \frac{n^{n+\frac12}}{e^n} < n! < c_2 \cdot \frac{n^{n+\frac12}}{e^n} \)

for all \(\displaystyle n\in \N\).

Give me another random problem!