SERIES.] A L G 3.2 1 . 2 776 1.2 E B It A Ex. 3. 561 The clianco of drawing a white and a black is (Art. 139, Prop 3), 3.4 _4 776 ~7 * 1.2 To find the chance of drawing at least a white ball, we may remark that it is the same as the chance of not draw ing two black balls, i.e., certainty the chance of drawing two black balls. Now the chance of drawing two black balls is 4_._3 172 2 7~U 7 1 .2 . . the chance of drawing at least one white ball is 2 5 1-77- SECT. XIX. ON SERIES IN GENERAL; THEIR SUMMATION AND CONVERGENCE. 141. Certain series, from their very appearance, indicate that they are really the sums or differences of two other scries. From this circumstance their sum may frequently be determined, as in the following examples: 1 n + l 142. The sum of a series may often be easily found by the method of increments or differences, and this method is especially adapted to the summation of integral series, such as the squares of the natural numbers. We shall exhibit one or two illustrations only. If we write S n = n(n+ 1), we have Hence conversely, and dividing by 2; if ,, . then will Similarly, if c, S .. = S. +1 -S. = 2) . . . (n + r-1), then will S = This last conclusion, of course, assumes that S,, is when n is 0. If it be otherwise, some numerical constant, easy of determination, will have to be added. Ex. 2. Here S nf , - S n = (n + I ) 2 = (n + 1 ) (n + 2) - (n + 1 ) ; Q _ n(n + 1) (n + 2) (?H-1) n(n+T) (2n + 1) " o o -7. Let = (n + 1) ( + 2) (n + 3) (i + 4) + A(?i + 1) (?i + 2) (n + 3) Dividing by ?i+l, and proceeding as in Art. 33, we get A = - 6 , B = 7 , = - 1 . S = g ( + 1 ) + 2 ) (n + 3) (n + 4) - 30 3n(n+ 1) - 1 > 1.2 " "2. 3 Let x 1- - + . 1 1 11 x- =- + -+ n 1 7i+l 2 3 .*. by subtraction, n+l 1 1 1 1
n+l)
I 1.2 2.3 1 n (+!) (9;i the Convergency aitd Divergency of Infinite Series. 143. Def. If the limit to which the sum of a series approaches, as the number of terms increases, is finite, the series is a converging series; if otherwise, diverging. For example, the sum of the series 1 + r + r 2 + . . . to n 1 r" terms is (Art. 52), which, when r is less than 1, l r ^ approaches to - , in which case the scries is a con verging series. Prop. 1. It is necessary and sufficient for convergency that the remaining terms after the nth have zero for their limit, both individually and collectively, as n increases. It is obviously necessary and sufficient for convergency that the sum of the series after the nth term shall have as its limit; and consequently, when all the terms of the series are positive, the same must be true of each indi vidual term. But when the terms are alternately positive and negative, though it is necessary for convergency that the sum of the consecutive terms with their proper signs should have as its limit, this is not sufficient ; for, were it so, the sum to n terms would depend on whether n is even or odd. Ex. 1. 1+^ + -,+ is n t a converging series ; for Z O although each term after the nth tends to as its limit the sum of n terms after the nth, viz., - --) + . . . - , which is greater than + + . . . -- to n terms, 2n 2/i 2rt 2n i.e., greater than -, does not tend to as its limit. A I .2 1.2.3 e (Art. 129), is convergent. The sum of the terms after the nth is 17 i + -!-+*,) n + l ) <uT . . . the expression for the limit of which as n increases is 0. Prop. 2. If the limit of the tli term is 0, and the terms continually diminish; then when the signs of the terms are alternately + and - , the series is convergent. Let j - MO + ?< 3 - etc., be the series; the terms after the nth (+ or ) make up the series of positive groups (+i - . +2 ) + (.+3 - +) + &c. But these terms may also be written u m+l - (u n+1 - n n+3 ) &c., which, since the whole group is positive, must
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