Solution
= Solution
Under the hypotheses of <Taylor's theorem with Lagrange remainder>, for example $f\in C^{n+1}$ on an interval containing $x$ and $x+h$, there is a point $\xi$ between them such that
$$
\boxed{f(x+h)=\sum_{r=0}^n\frac{f^{(r)}(x)}{r!}h^r+\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}.}
$$
The <Lagrange remainder> makes this an exact equality, rather than just a truncated <Taylor polynomial>. For $h=0$ the equality is immediate.