Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 15 / a

The first $n$ terms of Taylor's theorem, with the Taylor remainder in mean-value form, arewhere $\xi$ lies between $a$ and $x$ for $x\ne a$. Sufficient hypotheses are continuous derivatives through order $n-1$ on the interval between the endpoints and an $n$th derivative in its interior. At $x=a$ the remainder is zero. The finite Taylor polynomial ends at degree $n-1$; this identity does not require convergence of an infinite Taylor series.