Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/252

Since, however, we see at once that

$\left[{\frac {1}{1-x}}\right]_{x=.5}=2,$

there is no necessity for the above discussion, except for purposes of illustration. As a matter of fact the process of computing the value of a function from an equivalent series into which it has been expanded is of the greatest practical importance, the values of the elementary transcendental functions such as the sine, cosine, logarithm, etc., being computed most simply in this way.

So far we have learned how to expand only a few special forms into series ; we shall now consider a method of expansion applicable to an extensive and important class of functions and called Taylor's Theorem.

144. Taylor's Theorem^[1] and Taylor's Series. Replacing $b$ by $x$ in (E), §107, the extended theorem of the mean takes on the form

(61)	$f(x)=f(a)+{\frac {\left(x-a\right)}{1!}}f'\left(a\right)+{\frac {\left(x-a\right)}{1!}}f'\left(a\right)+{\frac {\left(x-a\right)^{3}}{3!}}f'''\left(a\right)+\cdots$
	$+{\frac {\left(x-a\right)^{n-1}}{\left(n-1\right)!}}f^{n-1}\left(a\right)+{\frac {\left(x-a\right)^{n}}{n!}}f^{n}\left(x_{1}\right),$

where $x_{1}$ lies between $a$ and $x$. (61), which is one of the most far reaching theorems in the Calculus, is called Taylor's Theorem. We see that it expresses $f(x)$ as the sum of a finite series in $(x-a)$.

The last term in (61), namely ${\frac {\left(x-a\right)^{n}}{n!}}f^{\left(n\right)}\left(x_{1}\right)$, is sometimes called the remainder in Taylor's Theorem after n terms. If this remainder converges toward zero as the number of terms increases without limit, then the right-hand side of (61) becomes an infinite power series called Taylor's Series^[2]. In that case we may write (61) in the form

(62)	$f(x)=f(a)+{\frac {\left(x-a\right)}{1!}}f'\left(a\right)+{\frac {\left(x-a\right)}{1!}}f'\left(a\right)+{\frac {\left(x-a\right)^{3}}{3!}}f'''\left(a\right)+\cdots ,$

and we say that the function has been expanded into a Taylor's Series. For all values of $x$ for which the remainder approaches zero as $n$ increases without limit, this series converges and its sum gives the exact value of $f(x)$, because the difference (= the remainder) between the function and the sum of $n$ terms of the series approaches the limit zero (§15).

1. Also known as Taylor's Formula.
2. Published by [Brook Tyalor] (1685-1731) in his [Methods Incrmentorum], London, 1715.