Taylor's Series, and neglect to take account of all terms after it, the error committed by so doing is

hn 1.2.3...n

F" (x+0h);

(16)

and as lies between 0 and 1, the error lies between

hn {"(x) and F" (x + h)} ;

1.2.3...n

(17)

which is the same value as that before determined in Art. 72. Similarly, if we stop at the nth term of Maclaurin's Series, and do not take account of the terms after it, the error is

xn 1.2.3...n

F" (0x).

(18)

which is the same value as that before determined in Art. 58. The expressions (16) and (18) are called respectively the limits of Taylor's and Maclaurin's Theorems.

135.] The series given in Article 133 enables us to put the limit of Taylor's Series under another form which is sometimes convenient.

Let us represent the last term of (13) by $(a); so that

Let a be variable, and let us differentiate this; and we have

(x-a)"-1 1.2.3... (n-1)

F" (a) + '(a) = 0.

(21)

It is plain from (19) that (x) = 0; therefore, writing in (14) for F, we have

(a)+(x− a) ' {a +01 (x − a)} = 0,

0, being different from 0 in (19);

.. (a) = (x − a) p' {a + 01 (x −α)}. In (21) for a write a +01(x-a), and we have (x-a)-(1-0)"-1

1.2.3... (n-1)

(22)

F{a+01(x-a)} + {a + 01 (x− a)} = 0; (23)

whence, by elimination between (22) and (23),

and substituting in (13) this particular form of the remainder,

Substituting in this series a + h for x, and subsequently writing

The corresponding expression for the remainder, in Maclaurin's Series, is

(1-0)"-1" 1.2.3... (n-1)

Two examples of the remainders of series in these forms are subjoined.

Ex. 1. F(x) = a* ;

.'. F" (x) = a* (log a)" ;

and therefore the sum of all the terms after the nth in the expansion of a is

Ex. 2. F(x) = sin mx; .. F"(x) = m" sin (

therefore the sum of all the terms after the nth of sin m (x + h) is

136.] In these last Articles, as well as in Chapter IV, we have frequently used the expression "finite and continuous for all values of x between" certain limits; now although we cannot tell all the cases wherein these conditions are, and are not fulfilled, for a complete knowledge of all functions and all their properties would be necessary to our doing so, yet we do know certain cases where they are not satisfied; these we propose to discuss briefly. In them Taylor's and Maclaurin's Series are PRICE, VOL. I.

Ff

said to fail; which is surely an incorrect term, for we are endeavouring to bring all functions within a particular formula, which is true only of some; that is, we are trying to make that which is true only when certain conditions are satisfied, comprise all, whether such conditions are fulfilled or not. We shall omit the consideration of discontinuous functions, as such are excluded by their very forms, and confine our attention to those which for particular values of the variable become infinite, and on that account fail to satisfy the requisite conditions.

First let us consider Taylor's Series. To develope F(x+h) to n + 1 terms, and so that the last term of the series may comprise the sum of all the terms after the nth, and be

it is necessary that r(x), and all its derived-functions up to F(x) inclusively, should not be infinite for any value of x between a and +h. Now let the following effects of differentiation on functions be observed; (1) that all rational and integral

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functions of x, and all functions of the form "+cx¬m+exq are lowered one dimension by it; and that therefore, if n is positive and integral, the nth derived-function of a" is a constant, and the (n+1)th is zero; that however many times x-m is differentiated, the result is never zero, but that the negative exponent is increased; that if is a fraction proper or improper,

some derived-function of it, sooner or later, has a negative exponent, and therefore some power of a appears in the denominator; (2) that if e() is a factor of the original function, it is also a factor of every one of the successive derived-functions.

Excluding all consideration as to values outside those for which the series is applied, let a be a particular value of x within them; and suppose that when x = a, F" (x) is finite, but F+1(x) is infinite: then it is plain that some factor of the form x-a must, in passing from F"(x) to F"+1(a), have been introduced into the denominator; and therefore, as far as the above remarks go, there must have been in the original function a factor of the form (-a)+, where m is an integral and positive number, and is a proper fraction. And if the primitive

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and all its derived-functions are infinite, when xa, there must in the original function have been a factor of the form (x-a)-".

Thus in the former case, suppose that we have to expand

in which neither f(x) nor (x) involves factors of the form x-a; then all the derived-functions up to F(x) will be finite, when a = a, but the subsequent ones will be infinite; the expansion therefore must not be carried beyond the mth term; and the addition of

hm 1.2.3...m

will make the equation exact.

F(x+0h),

Suppose, for instance, that it is required to expand r(a+h), having given

F(x) = x2 + (x− a) sin x;

then a is the least value of x for which the function is considered, and a+h is the greatest; and

5

F′(x)

= 4x3 +

(x − a)a sin x + (x−a) § cos x,

2

15

x''(x) = 12x2 + + (x− a) ‡ sin x + 5 (x − a) ‡ cos x — (x — a) § sin x.

4

But if we form F""(x), it involves (x-a), which becomes infinite when x = a, and therefore fails to fulfil the conditions under which equation (8) has been determined: therefore, in this case,

F(x + h) = = F(x) + — F′(x) +
1

h
-

h2 1.2

F'(x+0h);

and substituting the specific values above given, and putting

x = a,

F(a+h)

15

+ (Oh) sin (a + 0h) +5 (0h)3 cos (a+0h) — (0h)3 sin (a + 0h)},

4

0 being a positive and proper fraction. On inspection it is plain that the first three terms of the series are correct terms of the expansion of (a + h), and that they might have been carried further; and that the last three express (-a) sin x, when x = a + h.

Now observing that the last three terms involve h, we have a good illustration of the reason why Taylor's Series "fails;" viz. because we are attempting to express by a formula which involves only integral powers of h, a function which in its development requires fractional powers. Another and perhaps better illustration is the following: Let us consider the case of (x2-a2), which is to be developed, when x = a + h; here then, when a = a, the original function = 0, and the firstderived is; but as the function may be written in the form (x − a) 3 (x + a), this becomes, when x = a+h, h3 (2a+h)*, the second factor of which may be developed in the ordinary way, and thereby the whole development will consist of terms

of the powers of h whose exponents are of the form n + being a whole number.

1

2'

n

Similarly, if the original function has factors of the form (x-a)-", that and all its derived-functions are infinite when x = a. Thus suppose that we have to develope by Taylor's Series, when ra+h,

this and all its derived-functions become infinite when x = a; but the function may be written in the form

1 h'

the first factor of which, when x = a + h, becomes and the

second fulfils the conditions of Taylor's Series and may be expanded in the ordinary way; but the resulting development will have at least one term involving a negative power of h, which indicates the cause of the failure, viz. that the function does not admit of development in ascending integral and positive powers of h, which alone is given by Taylor's Theorem.

Secondly, as to the failure of Maclaurin's Theorem. Two cases correspond to those of Taylor's Series above discussed. In the former, the original function contains factors of the form X ; and therefore, when these are differentiated more than m times, powers of a with negative indices are introduced; and these become infinite when x = 0. Although therefore the series