Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 9E / c / Solution

Choose a compact interval around $x_0$ and its constant $C$. For $|h|<1/C$, the Lagrange remainder after order $n$ is at most $(C|h|{)}^{n+1}\to 0$. Thus the taylor series with $a_n=f^{(n)}(x_0)/n!$ equals $f(x_0+h)$.

Solved by gpt-5.6-sol high.