And a third way is by Newton's bino mial theorem, which is an universal me thod that serves for all sorts of quanti ties, whether fractional or radical ones: and by this means the same root of the last given quantity becomes %/a' -9-c' 1.3 a2a 2.4;33 2.4.6a3 .&c. where the law of continuation is vi sible.
Hence it appears that the signs of the terms may be either all plus, or alter nately plus and minus, though they may be varied in many other ways. It also appears that the terms may be either continually smaller and smaller, or larger and larger, or else all equal. In the first case, therefore, the series is said to be a decreasing. one ; in the second case, an in creasing one ; and in the third case, an equal one. Also the first series is called a converging one, because that, by collect ing its terms successively, taking in al one term more, the successive sums approximate or converge to the value or sum of the whole infinite series. So in the series 1 1 1 1 1 1 Ti = = 1 the first term 3 is too little or below 1 which is the value or sum of the whole infinite series proposed; the sum of the 1 1 9 4 = first two terms 3 + is .4444, &c.
9 1 is also too little, but nearer to 7 or .5 than the former; and the sum of three 1 1 1 13 terms + + 27 27 is= .481481, &c. is nearer than the last, but still too little; and the sum of four terms 1 1 1 1 0 = .493827, &c. 4 + + + is 3 9 27 81 81 which is again nearer than the former, but still too little ; which is always the case when the terms are all positive. But when the converging series has its terms alternately positive and negative, then the successive sums are alternately too great and too little, though still ap proaching nearer and nearer to the final sum or value. Thus, in the series 1 1 n 1 1 1 27 = 3 + 1 4 3 § 1 the 1st term = .333, &c is too great; 1 1 9 two terms 3 = .222, &c. are too little ; 1 1 1 7 three terms 3 + .259259, &c. are too great.
1 1 1 1 four terms + 3 9 27 81 .246913, &c. are too great, and so on, al ternately too great and too small, but every succeeding sum still nearer than the former, or converging: In the second case, or when the terms grow larger and larger, the series is call ed a &rosins. one, because that, by col lecting the terms continually, the succes sive sums diverge, or go always further and further from the true value or radix of the series; being all too great when the terms are all positive, but alternately too great and too little when they are al ternately positive and negative. Thus, in the series 1 1 , n , A .
4 2 3 the first term + 1 is too great; two terms 1 2 = 1 are too little ; three terms 1 2 + 4 = + 3 are too great ; four terms 1 are too little; and so on continually, after the 2d term, diverging more and more from 1 the true value or radix but alternate ly too great and too little, or positive and negative. But the alternate sums would be always more and more too great if the terms were all positive, and always too little if negative.
But in the third case, or when the terms are all equal, the series of equals, with alternate sums, is called a neutral one because the successive sums, formed by a continual collection of the terms, are always at the same distance from the true value or radix, but alternately posi tive and negative, or too great and too little. Thus, in the series,
, , , , -r--r - 1+1 2 =a &c.
the first term is 1 is too great; two terms 1 1 = 0 are too little ; three terms 1 1 + 1 = 1 too great ; four terms 1 1 + 1 1 = 0 too little; and so on, continually, the successive sums being alternately 1 and 0, which are equally different from the true value, or 1 , 5 the one as much above it as the other below it.
A series may be terminated and ren dered finite, and accurately equal to the sum or value, by assuming the supple ment, after any particular term, and com bining it with the foregoing terms. So, 1 1 1 ' in the series 2 4 &c. which 16 is equal to s, and found by dividing 1 by 2 + 1, after the first term, 2, ofthe quo tient, the remainder is which, di vided by 2 + 1, or 3, gives 6for the supplement, which combined with the first term, gives 1 the true 2' 1 1 2 6 sum of the series. Again, after the first 1 1 ' two terms '2- -- 4 the remainder is + ' which, divided by the same divisor, 3, 1 gives for the supplement, and this 1 1 4 combined with those two terms 2 ' , 1 1 , 1 4 makes 1 1 -I- 3= 4:175 or the same sum or value as before. And, in general, by dividing 1 by a + c, there is obtained 1 1 c cn a + c a. a3 (01+1 en where .stopping the di an + (a + c)' vision at any term, as , the remain.
der after this term is -7-7 + , which, be ing i.iihd tk same divisor, a + c, +i gives --- for the supplement as an +1 (a + c) above The Law of Continuation."-A series being proposed, one of the chief spies thins concerning it is to find the law of its continuation. Indeed no universal rule can be given for this ; but it often hap pens, that the terms of the series, taken two and two, or three and three, or in greater numbers, have an obvious and simple relation, by which the series may be determined and produced indefinite ly. Thus, if 1 be divided by 1- x, the quotient will be a geometrical progres sion, pis. 1 x x' &c. where the succeeding terms are produced by the continual multiplication by x. In like manner, in other cases of division, other progressions are produced.
But, in most cases, the relation of the terms of a series is not constant, as it is in those that arise by division. Yet their relation often varies according to a cer tain law, which is sometimes obvious on inspection, and sometimes it is found by dividing the successive terms one by an other, &c. Thus, in the series 2 8 16 128 1+ T + 515 &c by dividing the 2d term by the 1st, the 3d by the 2d, the 4th by the 3d, and so on, the quotients will be 2 4 6 8 , s Tx, OLC. and, therefore, the terms may be conti nued indefinitely, by the successive mul tiplication by these fractions. Also in the following series 3, 5 , 35 1 x4, 6 40 128 1152 &c. by dividing the adjacent terms suc cessively by each other, the series of quotients is 1 9 25 49 , - r, - r, - am; or 6 20 42 72 1 . 1 3 3 5 . 5 7 . 7 k 2 . 3 4 6. 8 . 9 and, therefore, the terms of the series may be continued by the multiplication of these fractions.