Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 9 / Solution

For an $N+1$ times continuously differentiable function, Taylor's theorem giveswhere $\xi$ lies between $x_0$ and $x$. An infinite Taylor series represents the function when its remainder tends to zero; this holds locally for an analytic function, rather than for every smooth function automatically.