TITLE: $\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$
QUESTION [6 upvotes]: Why is 
$$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$$
is there an intuitive or graphical way to think about it?

REPLY [2 votes]: Consider that $\log(x)$ is an increasing function and that $\log(1)=0$, then look at the following plot:

Normally, if $f(x)$ is an increasing function, then, for $a,b\in\mathbb{Z}$,
$$
\sum_{k=a+1}^bf(k)\ge\int_a^bf(x)\,\mathrm{d}x
$$
but since $f(a)=\log(1)=0$, we can include $1$ in the summation.