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every case of the expansion of (a + x)

±

should be shewn not to exceed any finite, ratio; but the investigation of this property, he found, led so directly to the law of the coefficients in each case, that there was no advantage in deducing it separately, in order, afterwards, to form one general demonstration. He could not, however, proceed satisfactorily, in his inquiry into the nature of the series obtained by algebraic evolution, without the help of the Polynomial Theorem. This theorem, therefore, he was obliged previously to demonstrate; the demonstration of it followed, with the greatest ease, from the method which he had employed for the simple involution of the binomial; and it gave him, more than he at first expected to follow immediately from it, namely, the law of the coefficients of the expansion of (a + x). He has since found that he had been anticipated in this step by Dr. HUTTON.

Complete precision has, undoubtedly, been aimed at in the following proof; the reader will judge whether it has been attained. One objection, however, may probably be foreseen, namely, that some of the conclusions here drawn, may seem not to be arrived at without the help of induction. This term properly denotes the inferring some general proposition from observing that it is true in a multitude of separate instances; no necessary connexion being perceived between the instances themselves and the common pro

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perty. It is thus that the laws of motion are collected, and other axioms in Physics. Induction, in this, the proper sense of the word, is wholly inadmissible in abstract mathematical science. But there is nothing of this kind in the inference, drawn from a partial division of unity by 1-x, that the mth term of the quotient must be a-1; the connexion between the subject and the prædicate, the form, and the law of continuation, of the series, is intuitively known; and the mind is as fully satisfied of the truth of the assertion, as it can be of that of any other proposition in Euclid's Elements. The same may be said of the equations which determine the qth terms of the involution of a binomial, or a polynomial, raised to the m power. When a series admits of being continued without limit, as the expansion of (a±x)-" and (a ± x)**, only a finite number of terms can then be exhibited; but if, in all cases, the law of the formation of the terms be found for any two successive terms whatever, the pth and the (p + 1)th, it may be fairly concluded to obtain in them all; and this law being once determined, the series, may be continued to any number of terms whatever, without Induction.

ART. 1. DEF. The word Function in the following articles, is used to designate any algebraic

expression, containing one or more variable quantities, mixt, or not, with constant quantities; such an expression is called the Function of the variable quantity, or quantities, which it contains.

2. DEF. The Limit of a Function is a constant quantity, from which the function may be made to differ less than any other given quantity, but to which it can never be equal.

PROP. I.

3. Theorem. If, in a series of quantities, continued indefinitely, each term be the half of that immediately preceding it, any term whatever is greater than the sum of all the terms which follow it.

Let K be any term whatever of such a series;
K K K

therefore

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&c. are the succeeding

terms; but, by the common rule, investigated in the Elements of Algebra, 2 K is the limit of the

K K
2
4

progression K + + + &c. ad infinitum ;

that is, 2 K is greater than K +

K K

+ + &c. to 4

2

whatever number of terms the series be continued;

take K from both, and there remains K greater

K K

K

8

than + + + &c. ad infinitum.

2

4. COR. If each term of a series be less than the half of that immediately preceding it, any term whatever is greater than the sum of all the terms which follow it.

PROP. II.

5. Theorem. In a series of quantities, either finite or continued indefinitely, of the form A ± Bx + Cx2± Dr3 + &c., in which the value of x is arbitrary, and the coefficients A, B, C, &c. are constant quantities, none of which has to that preceding it a ratio greater than any ratio assignable, x may be taken such that any term shall be indefinitely greater than the sum of all the terms of the series which contain higher powers of x.

Let the constant coefficients A, B, C, &c. be supposed to be all positive, and to go on increasing from A, so that the second is greater than the first, the third than the second, and so on; and let RxP, SxP+1 be the two successive terms in which the ratio of the latter coefficient to the former is greatest; then, since this ratio, by the

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2.S

is a finite quantity, and,

therefore, also is a finite quantity; but the

R

value of x is arbitrary; let, then, a become less

R

than

25; therefore S. x is less than and

S. +1 less than

R
2

R

2

.x; that is, the term SP+ 1

is less than the half of the preceding term; but, by the supposition, no other coefficient has to that of the term immediately preceding it, so great a ratio as S has to R; wherefore, the second term is less than the half of the first, and every term in the series, after the first, is less than the half of the preceding term; therefore, (Art. 4.) any term of the series is greater than the sum of all the remaining terms; and, if this be the case, when the coefficients are supposed to be all positive, and to go on increasing from A, it will, a fortiori, be the case, when any other supposition is made, relative to the magnitudes, and the signs of the coefficients; wherefore x may always be so taken, as that any term of the series shall be greater than the sum of all the terms that follow it and, it is manifest, that by continually diminishing the arbitrary value of x, any term may be made to become indefinitely greater than the sum of all the following terms.

:

*All the coefficients, A, B, C, &c., of the series being supposed to have finite and constant values, the proposition may be proved, independently of Art. 4. in the following manner.

First, the value of x may be taken such, that the first term A of the series A + B x + Cx2+ &c. shall have to any other term, as 2 x2, a ratio greater than any assigned ratio, as that of r to unity,

For,

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