Determine whether \(\frac{1}{2}(L+R)\) underestimates or overestimates the value of the integral.
Sequences
Question
Show, graphically or otherwise, that if \(r\) is an integer greater than \(1\), then
\[\begin{equation*}
L = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dotsb + \frac{1}{r} < \int_1^r \frac{1}{x} \:dx < 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{r-1} = R.
\end{equation*}\]
