= Solution
The cubic <Taylor polynomial> about $a$ is
$$
\boxed{f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3.}
$$
If $f$ has a continuous fourth <derivative> near $a$, <Taylor's theorem> gives $f(x)$ equal to this <polynomial> plus $O((x-a)^4)$. With continuous <derivatives> only through third order, the corresponding remainder is $o((x-a)^3)$. An equality with an infinite <Taylor series> would require additional convergence information, rather than just the existence of these four terms.
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