= Solution
With the first $n$ terms counted from the constant term, <Taylor's theorem> gives
$$
\boxed{f(x)=\sum_{k=0}^{n-1}\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n(x),\qquad
R_n(x)=\frac{f^{(n)}(\xi)}{n!}(x-a)^n,}
$$
where $\xi$ 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.
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