Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 15 / a

For $f$ sufficiently differentiable on the interval between $a$ and $x$, Taylor's theorem with its first $n$ terms readswhere $\xi$ lies between $a$ and $x$. This is the Taylor theorem with Lagrange remainder. For example, $|f^{(n)}|\le M$ on that interval gives $|R_n(x)|\le M|x-a{|}^n/n!$. The finite formula does not assume that an infinite Taylor series converges to the function; that further conclusion requires the remainders to tend to zero.