If a function \(f\) is differentiable through order \(n+1\) in an interval \(I\) containing \(c\), then for each \(x \in I\),
\[f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!} (x-c)^2 + \ldots + \frac{f^{(n)}}{n!}(x-c)^n + R_n(x)\] where the remainder term \(R_n(x)\) is \[R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}\] for some \(z\) between \(x\) and \(c\).