Prove that $2 < e < 4$ using upper and lower Riemann sums and the definition of $\ln{x}$
I think I understand the concept of what I need to do, but I am having some trouble implementing a solution. I guess this would be equivalent to showing that $\ln(2) < 1 < \ln(4)$ since the $\ln$ function is increasing.
What I'm not sure about is how I use the definition of $\ln(x)$ in the Riemann sum. I tried this:
$\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n}\int_1^{\frac{k}{n}} \frac{1}{t}dt$
I wasn't sure how to check the value at each point in order to prove my inequalities. How am I supposed to be doing this?