I'm trying to understand how the maximum error when using Riemann sums is equal to the difference between the overestimate and underestimate. I'm presented with the following equation:
\begin{align} |\text{overestimate} - \text{underestimate}| & = |\sum_{i=1}^{n} f(x_{i})\Delta x - \sum_{i=0}^{n-1} f(x_{i})\Delta x | \\\\ & = |f(x_{n}) - f(x_{0})|\Delta x \end{align}
I get that I can factor out the $\Delta x$ from both summations, but how is it possible that the sum from $i=1$ to $n$ on $f(x)$ is equal to $f(x_{n})$? This equation assumes that I have a fixed $n$.
If you take out the $\Delta x$ then the summation become $f(x_i) -f(x_i-_1)$ summing over which gives you the required expression. This is simply because the terms in the sum get cancelled out.