148. Taylor’s Series

Suppose that $f (x)$ is a function all of whose differential coefficients are continuous in an interval $[a - η, a + η]$ surrounding the point $x = a$ . Then, if $h$ is numerically less than $η$ , we have $f (a + h) = f (a) + h f^{'} (a) + \dots + \frac{h^{n - 1}}{(n - 1)!} f^{(n - 1)} (a) + \frac{h^{n}}{n!} f^{(n)} (a + θ_{n} h),$ where $0 < θ_{n} < 1$ , for all values of $n$ . Or, if $S_{n} = \sum_{0}^{n - 1} \frac{h^{ν}}{ν!} f^{(ν)} (a), R_{n} = \frac{h^{n}}{n!} f^{(n)} (a + θ_{n} h),$ 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 $f (a + h) = lim_{n \to \infty} S_{n} = f (a) + h f^{'} (a) + \frac{h^{2}}{2!} f ” (a) + \dots .$

This expansion of $f (a + h)$ is known as Maclaurin’s Series. When $a = 0$ the formula reduces to $f (h) = f (0) + h f^{'} (0) + \frac{h^{2}}{2!} f ” (0) + \dots,$ 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.

Example LVI

1. Let $f (x) = \sin x$ . Then all the derivatives of $f (x)$ are continuous for all values of $x$ . Also $| f^{n} (x) | \leq 1$ for all values of $x$ and $n$ . Hence in this case $| R_{n} | \leq h^{n} / n!$ , which tends to zero as $n \to \infty$ (Ex. XXVII 12) whatever value $h$ may have. It follows that $\sin (x + h) = \sin x + h \cos x - \frac{h^{2}}{2!} \sin x - \frac{h^{3}}{3!} \cos x + \frac{h^{4}}{4!} \sin x + \dots,$ for all values of $x$ and $h$ . In particular $\sin h = h - \frac{h^{3}}{3!} + \frac{h^{5}}{5!} - \dots,$ for all values of $h$ . Similarly we can prove that $\cos (x + h) = \cos x - h \sin x - \frac{h^{2}}{2!} \cos x + \frac{h^{3}}{3!} \sin x + \dots, \cos h = 1 - \frac{h^{2}}{2!} + \frac{h^{4}}{4!} - \dots .$

2. The Binomial Series. Let $f (x) = (1 + x)^{m}$ , where $m$ is any rational number, positive or negative. Then $f^{(n)} (x) = m (m - 1) \dots (m - n + 1) (1 + x)^{m - n}$ and Maclaurin’s Series takes the form $(1 + x)^{m} = 1 + (\binom{m}{1}) x + (\binom{m}{2}) x^{2} + \dots .$

When $m$ is a positive integer the series terminates, and we obtain the ordinary formula for the Binomial Theorem with a positive integral exponent. In the general case $R_{n} = \frac{x^{n}}{n!} f^{(n)} (θ_{n} x) = (\binom{m}{n}) x^{n} (1 + θ_{n} x)^{m - n},$ and in order to show that Maclaurin’s Series really represents $(1 + x)^{m}$ for any range of values of $x$ when $m$ is not a positive integer, we must show that $R_{n} \to 0$ for every value of $x$ in that range. This is so in fact if $- 1 < x < 1$ , and may be proved, when $0 \leq x < 1$ , by means of the expression given above for $R_{n}$ , since $(1 + θ_{n} x)^{m - n} < 1$ if $n > m$ , and $(\binom{m}{n}) x^{n} \to 0$ as $n \to \infty$ (Ex. XXVII. 13). But a difficulty arises if $- 1 < x < 0$ , since $1 + θ_{n} x < 1$ and $(1 + θ_{n} x)^{m - n} > 1$ if $n > m$ ; knowing only that $0 < θ_{n} < 1$ , we cannot be assured that $1 + θ_{n} x$ is not quite small and $(1 + θ_{n} x)^{m - n}$ quite large.

In fact, in order to prove the Binomial Theorem by means of Taylor’s Theorem, we need some different form for $R_{n}$ , such as will be given later (§ 162).

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147. Taylor’s Theorem

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149. Applications of Taylor’s Theorem to maxima and minima

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