# Math Help - A proof involving logs

1. ## A proof involving logs

Can anyone show that:

log(N-1)! $\leq$ $\int_1^N \! log(x) \, dx$ $\leq$ log(N)!

2. $\log (n!) = \log n(n-1)(n-2)\cdots2\cdot1 = \log n + \log (n-1) + \cdots + \log 2$. This is a Riemann sum for the integral $\int \log x$. Can you show that the Riemann sum on the left of your inequality is less than the integral?

3. What Maddas is doing is setting up a Riemann sum for $\int_1^N log(x)dx$ using the integers as the "break points"- that is $\delta x= 1$. Since log(x) is an increasing function, you get the "lower sum" using the left endpoints of each interval and the "upper sum" using the right endpoints.

4. So i would say that the one on the LHS is the Riemann integral for the limit 1 to N-1....

So the middle bit of the inequality is the Riemann integral of the LHS + log(N)?