Applications of the Binomial Theorem. 240. If in the formula a3 (x+a)= 3 The fifth term can be found by multiplying the fourth by and by a √x+a= ...... then changing the sign of the result, and so on. 3n-1 4η 241. REMARK. When the terms of a series go on decreasing in value, the series is called a decreasing or converging series; and when they go on increasing in value, it is called a diverging series. In a converging series the greater number of terms we take in the series, the nearer will we approximate to the true value of the proposed series. When the terms of the series are alternately positive and negative, we can, by taking a given number of terms, determine the degree of approximation. وه . . For, let a-b+c-d+e-f+ &c. be a decreasing series, b, c, d. being positive quantities, and let a denote the number represented by this series. .. The numerical value of x is contained between any two consecutive sums of the terms of the series. For take any two consecutive sums, a-b+c-d+e-f, and a-b+c-d+e-f+g. In the first, the terms which follow -f, are g-h, +k-l+... but since the series is decreasing, the partial differences g-h, k-l,.... are positive numbers; therefore, in order to obtain the complete value of x, a certain positive number must be added to the sum a-b+c-d+e-f. Hence we have a-b+c-d+e-f<x. In the second series, the terms which follow +g are -h+k, -l+m.... Now, the partial differences -h+k, -l+m..., are negative; therefore, in order to obtain the sum of the series, a negative quantity must be added to a-b+c-d+e-f+g, or, in other words, it is necessary to diminish it. Consequently a-b+c-d+e-f+g>x. Therefore, a is comprehended between these two sums. The difference between these two sums is equal to g. But since x is comprised between them, their difference g must be greater than the difference between a and either of them; hence, the error com. mitted by taking a certain number of terms, a-b+c-d+e-f, for the value of x, is numerically less than the following term. 242. The binomial formula also serves to develop algebraic expressions into series. 1 Take for example, the expression 1, we have In the binomial formula, make m=-1, x=1, and a=-z, it be or, performing the operations, and observing that each term is composed of an even number of factors affected with the sign -, 1 1-z (1-x)=__=1+z+z+z+z*+*+.... The same result will be obtained by applying the rule for divi 2 Again, take the expression (1-2)3 , or 2(1-2)-3. or 2(1-2)=2(1+3z+6z2+10z3+15z*+....) To develop the expression V2z-z2 which reduces to √22(1-2), we first find (1-)=1+(-)+(-)+... 2 2 1 1 =1-0-36 V 2z 5 -23 648 2 5 2 hence 22-2=22(1-2-3-648-, &c.) 1. To find the value of finite series. Method of Indeterminate Co-efficients. Recurring Series. 243. Algebraists have invented another method of developing algebraic expressions into series, which is in general, more simple than those we have just considered, and more extensive in its appli. cations, as it can be applied to algebraic expressions of any nature whatever. Before considering this method, it will be necessary to explain what is meant by the term function. Let a=b+c. In this equation, a, b and c, mutually depend on each other for their values. For, a=b+c, b=a-c, and c=a-b. The quantity a is said to be a function of b and c, b a function of a and c, and ca function of a and b. And generally, when one quantity depends on others for its value, it is said to be a function of those quantities on which it depends a In order to give some idea of this method of development, we will suppose it is required to develop the expression into a se. a'+bx ries arranged according to the ascending powers of x. It is plain a that the expression can be developed; for reduces to a+b'x a(a'+b'x); and by applying the binomial formula to it, we should evidently obtain a series of terms arranged according to the ascending powers of x. We may therefore assume a a+b'x =A+Bx+Cx2+Dx3+Ex2+Fx2+....(1) the co-efficients A, B, C, D, ... being functions of a, a', b', but independent of x, it is required to determine these co-efficients, which are called indeterminate co-efficients. For this purpose, multiply both members of the equation (1) by a'+b'x; arranging the result with reference to the powers of Now if the values of A, B, C, D,... were determined, the equation (1) would be satisfied by any value given to x; this must therefore be the case also in the equation (2). But by supposing x=0, this equation becomes, a a'' A being equal to when x=0, this must be the value of it when x is any quantity whatever, since A is independent of æ by hypothesis; therefore whatever may be the value of x, the equation (2) This equation being also satisfied by any value for æ, by making |