g(x)=\left\{
\begin{array}{ll}
\frac{f(x)-\sum_{k=0}^n\frac{f^{(k)}(0)}{k!}x^k}{x^{n+1}}, & x\neq 0,
\vspace{0.5em}\\
\frac{f^{(n+1)}(0)}{(n+1)!}, & x=0,
\end{array}
\right.