Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 30A / a / ii / Solution

For every $N$,Integrating the finite sum givesFor every $\epsilon >0$, sufficiently close to $x_0$ one has $|r_N(\xi )|\le \epsilon |\xi -x_0{|}^N$, so the last integral is $o(|x-x_0{|}^{N+1})$. This proves the integrated expansion.