= Solution
The first four terms form the cubic <Taylor polynomial> about $x_0$:
$$
\boxed{f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2
+\frac{f^{(3)}(x_0)}{3!}(x-x_0)^3.}
$$
With a continuous fourth <derivative> nearby, <Taylor's theorem> gives a remainder $O((x-x_0)^4)$. Continuous <derivatives> through third order suffice for the weaker remainder $o((x-x_0)^3)$. These are local approximations; equality with an infinite <Taylor series> requires additional convergence information.
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