Suppose that \(f(x)\) is a function all of whose differential coefficients are continuous in an interval \({[a – \eta, a + \eta]}\) surrounding the point \(x = a\). Then, if \(h\) is numerically less than \(\eta\), we have \[f(a + h) = f(a) + hf'(a) + \dots + \frac{h^{n-1}}{(n – 1)!} f^{(n-1)}(a) + \frac{h^{n}}{n!} f^{(n)}(a + \theta_{n} h),\] where \(0 < \theta_{n} < 1\), for all values of \(n\). Or, if \[S_{n} = \sum_{0}^{n-1} \frac{h^{\nu}}{\nu!} f^{(\nu)}(a),\quad R_{n} = \frac{h^{n}}{n!} f^{(n)}(a + \theta_{n} h),\] we have \[f(a + h) – S_{n} = R_{n}.\]

Now let us suppose, in addition, that we can prove that \(R_{n} \to 0\) as \(n \to \infty\). Then \[f(a + h) = \lim_{n\to\infty} S_{n} = f(a) + hf'(a) + \frac{h^{2}}{2!} f”(a) + \dots.\]

This expansion of \(f(a + h)\) is known as **Maclaurin’s Series.** When \(a = 0\) the formula reduces to \[f(h) = f(0) + hf'(0) + \frac{h^{2}}{2!} f”(0) + \dots,\] which is known as . The function \(R_{n}\) is known as **Lagrange’s form of the remainder**.

The reader should be careful to guard himself against supposing that the continuity of all the derivatives of \(f(x)\) is a sufficient condition for the validity of Taylor’s series. A direct discussion of the behaviour of \(R_{n}\) is always essential.

$\leftarrow$ 147. Taylor’s Theorem | Main Page | 149. Applications of Taylor’s Theorem to maxima and minima $\rightarrow$ |