But, if at first we write this fraction under the form Kam-n, and that we put a=0, we find that it becomes KX0m-n, which is 0 for m>n; in case of m<n, of m=n K K ―d, we shall have (Art. 86), which is equal to infi Od 0 ty, as has been already observed; finally, for m=n, we can divide above and below by am, and the fraction is reduced to K,, which is a finite quantity. 167. If we suppose, in the fraction (Art. 163), a=2, we find 1 1-2 =1+2+4+8+16+32+64+, &c., which at first sight will appear absurd. But it must be remarked, that if we wish to stop at any term of the above series, we cannot do so without joining the fraction which remains. Suppose, for example, we were to stop at 64; after having written 1+2+4+8+16+32+64, we must join the fraction 128 1-2 or 128 or -128; we shall therefore have for the complete quotient 127-128, that is in fact —1. Here, however far the fractional terin may be extended, its numerical value, which is negative, will always surpass, by a unit, that of the integral part, so that this is totally destroyed; and as in the hypothesis of a>1, we shall always subtract more than what we will add, we shall never meet with the 1 result 168. These are the considerations which are necessary when we assume for a numbers greater than unity; but if we now suppose a less than 1, the whole becomes more in 1 telligible; for example, let a=, and we shall have a 1 =2, which will also be equal to the following se 64 ries, 1+1+1+1+iotaatoit, &c., to infinity (Art. 163). Now, if we take only two terms of the series, we shall have 14, and it wants of being equal to 2; if we take three terms, it wants 1, for the sum is 13; if we take four terms, we have 13, and the deficiency is only ! 1 Therefore, we see very clearly that the more terms of the quotient we take, the less the difference becomes; and that, consequently, if we continue to take successive portions of this series, the differences between those consecutive sums and the fraction=2, decrease, and end by becoming less than any given number, however small it may be. The number 2 is therefore still a limit, according to the acceptation of this word. Now, it may be observed, that if the preceding series be continued to infinity, there will be no difference at all between its sum and the value of the fraction 1 or 2. 169. A limit, according to the notion of the ancients, is some fixed quantity, to which another of variable magnitude can never become equal, though, in the course of its variation, it may approach nearer to it than any difference that can be assigned; always supposing that the change, which the variable quantity undergoes, is one of continued increase, or continued diminution. Such, for example, is the area of a circle, with regard to the areas of the circumscribed and inscribed polygons; for, by increasing the number of sides of these figures, their difference may be made less than any assigned area, however small ; and since the circle is necessarily less than the first, and greater than the second, it must differ from either of them by a quantity less than that by which they differ from each other. The circle will thus answer all the conditions of a limit, which is included in the above definition. 170. The preceding considerations are very proper to define the nature of the word limit; but as Algebra, which is the subject we are treating of here, needs no foreign aid to demonstrate its principles, it is necessary, therefore, to explain the nature of the word limit, by the consideration of algebraic expressions. For this purpose, let, in the first place, the very simple fraction be in which we suppose that x may be positive, and augmented indefinitely; in dividing both terms of this fraction by z, the result, + evidently shows that x+a' a a a the function remains always less than a, but that it approaches continually to a, since that the part a, of its denominator, diminishes more and more, and can be reduced to such a degree of smallness as we would wish. 171. The difference between a and the proposed fraction be ax x+a a2 becomes so much ing in'general expressed by a smaller, according as x is larger, and can be rendered less than any given magnitude, however small it may be ; so that the proposed fraction can approach to a as near as we would wish a is therefore the limit of the fraction ax x+a' relative ly to the indefinite augmentation which x can receive. It is in the characters which we have just expressed, that the true acceptation, which we must give to the word limit, consists, in order to comprehend every thing which can relate to it. 172. If we had remarked in the preceding example, that by carrying on, as far as we would wish, the augmentation of x, we could never regard, as nothing, the fraction ; therefore a2. x+a we would reasonably conclude, that the fraction 2 α it would approach indefinitely to the limit a, could never attain a, and, consequently, cannot surpass it; but it would be wrong to insert this circumstance as a condition in the general definition of the word limit; we would thereby exclude the ratios of vanishing quantities, ratios whose existence is incontestable, and from which we derive much in analysis. 173. In fact, when we compare the functions ax and ax+ x2, we find that their ratio, reduced to its most simple expression, is and that it approaches nearer and nearer to unity, according as x diminishes. It becomes exactly 1, when x=0; but the quantities ax and ax+x2, which are then rigorously nothing, can they have a determinate ratio? This is what appears difficult to conceive; and we cannot give a clear idea of it but by presenting the quantity 1 as a limit to which the ratio of the functions ax and ax+x2 can approach as near as we would wish, since the difference, 1 a+ x a a+x XC can be rendered less than any assignable magnitude, a+x however small this magnitude may be. On the other hand, the ratio, α of the quantities ar a+x' and ax+x2 can not only attain unity when we make x=0, but Eurpass it when we suppose a negative, since it becomes then α This a quantity which is greater than 1, when x<α. circumstance appears not at all contrary to the idea of limit; for we can regard the value 1, which answers to x=0, as a term towards which the ratio of the functions ax and ax+x2 tends, by the diminutions of the values of x, whether positive or negative. For further illustrations of the word limit, and what is meant by infinity, and infinitely small quantities or in. finitessimals, the intelligent reader is referred to LACROIX'S Introduction to the Traité du Calcul Differentiel et du Calcul Integral, 4to. where these subjects are clearly elucidated. 174. Now, let a=1, in the fraction have 1 1 , and we shall =3=1+}+}+&{}+&+is+,&c. If we take two terms, we find 1+1, and the difference ; three terms give 1+3, the error =; for four terms the error is no more than Since, therefore, the error always becomes three times less, it tends toward zero, which it cannot attain, and the sum tends toward 2, which is the limit. 175. Again, let us take a=, and we shall have 6 : 1 =3 1-3 =1+2+3+3+1;+32+, &c. ; here, in the first place, the sum of two terms, which is 1+3, is less than 3 by 1+; taking three terms, which make 21, the error is ; for four terms, whose sum is 21. the error is 14. 1 16 `176. Finally, for a=4, we find,__=1+1=1+++is+o's ++, &c.; the first two terms are equal to 14, which gives for the error; and taking one term more, we shall have only an error of 177. From the preceding considerations we may readily conclude, that any fraction having a compound denominator may be converted into an infinite series by the following rule; and if the denominator be a simple quantity it may be divided into two or more parts. RULE. Divide the numerator by the denominator, as in the division of integral quantities, and the operation continued as far as may be thought necessary, will give the series required. Ex. 1. It is required to reduce ax ax into an infinite series. The terms in the quotient are found thus; dividing the first remainder x2, by a, the first term of the divisor α--x, we shall have for the second term of the quotient, because - α the division can be only indicated; multiplying the divisor by x2 α 23 a and subtracting the product from x2, the remainder is x3 a2 ; again, dividing this remainder by a, the result will be which is the third term in the quotient; and, in like manner, we might continue the operation as far as we please : But the law of continuation is evident, because the powers of x increase by unity in each successive term of the quotient, and the powers of a increase by unity in the denominator of each of the terms after the first. And the sum of the terms infinitely continued is said to be equal to the original fraction ax Thus we say that the numerical fraction, when reduced to a decimal, is equal to .6666, &c., continued to infinity. Ex. 2. It is required to convert ries. α into an infinite se |