§ VII. RESOLUTION OF ALGEBRAIC FRACTIONS OR QUOTIENTS INTO INFINITE SERIES. 160. An infinite series is a continued rank, or progression of quantities, connected together by the signs or; and usually proceeds according to some regular, or determined law. Thus, +++++++, &c. 10 In the first of which, the several terms are the reciprocals of the odd numbers 1, 3, 5, 7, &c. ; and in the latter the reciprocals of the even numbers, 2, 4, 6, 8, &c., with alternate signs. 161. We have already observed (Art. 96), that if the first or leading term of the remainder, in the division of algebraic quantities, be not divisible by the divisor, the operation might be considered as terminated; or, which is the same, that the integral part of the quotient has been obtained. And, it has also been remarked, (Art. 89), that the division of the remainder by the divisor can be only indicated, or expressed, by a fraction: thus, for example, if we have to divide aoby a+1, we write for the quotient : This, however, does not prevent us from attempting the division according to the rules that have been given, nor from continuing it as far as we please, and we shall thus not fail to find the true quotient, though under different forms. 1 a+1 162. To prove this, let us actually divide a° or 1, by 1--a, thus ; Now, by considering the first of these formulæ, which is 1+ have 1+ a 1-a a 1-a+a 1 1-a 1 a 1 -O 1-α 1 α If we follow the same process with regard to the second expression, that is to say, if we reduce the integral part 1+a to the same denominator, 1-a, we shall have the fraction 1-a2 1- -a to which if we add In the third formula of the quotient, the integers 1+a+a reduced to the denominator 1-a make Therefore each of these formulæ is in fact the value of the proposed fraction 1 163. This being the case, we may continue the series as far as we please, without being under the necessity of performing any more calculations; by observing, in the first place, that each of these formulæ is composed of an integral part which is the sum of the successive powers of a, beginning with a°1 inclusively; Secondly, of a fraction which has always for the denominator 1-a, and for the numerator the letter a, with an exponent greater, by unity, than that of the same letter in the last term of the integral part. This constant formation of the successive formulæ, is what Analysts call a law. And the manner of deducing general laws by the consideration of certain particular cases, is usually called induction; which, though not a strict method of proof, says LAPLACE, has been the source of almost all the discoveries that have hitherto been made, both in analysis and physics, of which all the phenomena are the mathematical results of a small number of invariable laws. It is thus that NEWTON, by following the law of the numeral coefficients, in the square, the cube, the fourth power, &c. of a binomial, arrived soon at the general law that bears his name, and which will be demonstrated in one of the following Chapters: This Geometer has carefully added, that in following this mode of investigation, we must not generalize too hastily; as it often happens, that a law, which appears to take place in the first part of a process, is not found to hold good throughout. Thus, in the simple instance of reto a decimal, its equivalent value is 17174949, &c., of which the real, repeating period is 49, and not 17, as might, at first, be imagined. ducing 531251 3093750 164. From what has been observed with regard to the successive quotients, (Art. 162), we can, in general, put 1 .1 an+1 a"+I 1-a n being a whole positive number, which augmented by unity, gives the place of the term. In fact, making n=3, a" becomes a3, which is the fourth term of the quotient; for n=4, a" becomes a1, which is the fifth term. But as nothing hinders us from removing indefinitely the fractional term which terminates the series, that is, of adding always a term to the integral part; so that we might still go on without end; for which reason it may be said that the proposed fraction has been resolved into an infinite series; which is, 1+a+ a2+a3+a1+a3+a® +a2+a®+a+a1o+a11+a +,&c. to infinity: and there are sufficient grounds to maintain that the value of this infinite series is 12 9 ·=1+a+a2+a3 +a1+; &c. 11 165. What has just been observed may at first appear strange; but the consideration of some particular cases will make it easily understood. Let us suppose, in the first place, a=1; the general quotient above will become a particular quotient corresponding to the fraction series taken indefinitely, shall be 1 1 The 1-1 1+1+1+1+1+1+,&c.. |