Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 15 / a

With the first $n$ terms counted from the constant term, Taylor's theorem giveswhere $\xi$ lies between $a$ and $x$. Continuous derivatives through order $n$ on the joining interval are a sufficient hypothesis for this Taylor theorem with Lagrange remainder. The Taylor remainder measures the error of the finite Taylor polynomial; convergence of the corresponding infinite Taylor series is a separate question.