This leads us to enquire under what conditions the product of two infinite convergent series is also convergent. *308. Let us denote the two infinite series а+а ̧x+а ̧x2 +a ̧ã3 + ... +α x2 + 2n 2n If we multiply these series together we obtain a result of the form ab+(a ̧b+ab) x + (a ̧b ̧ + a ̧b ̧ +ɑ ̧3 ̧) x2 + · ... Suppose this series to be continued to infinity and let us denote it by C; then we have to examine under what conditions C may be regarded as the true arithmetical equivalent of the product AB. First suppose that all the terms in A and B are positive. Let Aan B. C2, denote the series formed by taking the first 2n + 1 terms of A, B, C respectively. 2n 2n 2n 2n If we multiply together the two series A, B. the coefficient of each power of x in their product is equal to the coefficient of the like power of x in C as far as the term x2; but in A„B ̧ there are terms containing powers of a higher than a", whilst x2 is the highest power of x in C1; hence 2n 2n 2n If we form the product AB, the last term is ab2; but C includes all the terms in the product and some other terms besides; hence 2n C>A B. 2n Thus C is intermediate in value between AB, and AB2 whatever be the value of n. Let A and B be convergent series; put AA-X, BB - Y, 2n where X and Y are the remainders after n terms of the series have been taken; then when n is infinite X and Y are both indefinitely small. .. AB (A-X) (B-Y) = AB-BX-AY+XY; = therefore the limit of AB is AB, since A and B are both finite. " n must be equal to AB Therefore C which is the limit of C2 since it lies between the limits of AB and A, B, 2n 2n Next suppose the terms in A and B are not all of the same sign. 2n 2n > n 12 In this case the inequalities ABC AB are not necessarily true, and we cannot reason as in the former case. Let us denote the aggregates of the positive terms in the two series by P, P' respectively, and the aggregates of the negative terms by N, N'; so that A=P-N, B= P'-N'. Then if each of the expressions P, P', N, N' represents a convergent series, the equation AB=PP' - NP'— PN' + NN', has a meaning perfectly intelligible, for each of the expressions PP', NP', PN', NN' is a convergent series, by the former part of the proposition; and thus the product of the two series A and B is a convergent series. Hence the product of two series will be convergent provided that the sum of all the terms of the same sign in each is a convergent series. But if each of the expressions P, N, P', N' represents a divergent series (as in the preceding article, where also P = P and N'N), then all the expressions PP', NP', PN', NN' are divergent series. When this is the case, a careful investigation is necessary in each particular example in order to ascertain whether the product is convergent or not. *EXAMPLES. XXI. b. Find whether the following series are convergent or divergent: + 9. 1+1.2+ 1.2. (y+1) a (a+1)(a+2) B (B+1) (B+2) 10. 2 (log 2)+a3 (log 3)+a (log 4)+....... 11. 1+a+ 12. If Un Un +1 n*+ An-1+Bn*-2+ Cn2-3+ nk+ank-1+bnk-2+ cnk-3+. ... integer, shew that the series u1+u2++...... is convergent if A-a-1 is positive, and divergent if A-a-1 is negative or zero. CHAPTER XXII. UNDETERMINED COEFFICIENTS. 309. In Art. 230 of the Elementary Algebra, it was proved that if any rational integral function of x vanishes when x = a, it is divisible by x-a. [See also Art. 514. Cor.] be a rational integral function of x of n dimensions, which vanishes when x is equal to each of the unequal quantities Denote the function by f(x); then since f(x) is divisible by x-a1, we have f(x)=(x-α) (Për−1 + ..............), the quotient being of n-1 dimensions. Similarly, since f(x) is divisible by x-a,, we have ..). the quotient being of n − 2 dimensions; and Pox-2 + = (x -α) (Px-3+ ...... Proceeding in this way, we shall finally obtain after n di visions ƒ (x) = P。 (x -- α ̧) (x − α ̧) (x − α ̧) ................ (x — α„). ...... 310. If a rational integral function of n dimensions vanishes for more than n values of the variable, the coefficient of each power of the variable must be zero. Let the function be denoted by ƒ (x), where and suppose that f(x) vanishes when x is equal to each of the unequal values a1, α2, ɑz ...... a; then f(x)=P. (x − a) (x − α ̧) (x — α ̧) ...... (x — a„). Let c be another value of x which makes f(x) vanish; then since f(c) = 0, we have P. (c-a) (c-a) (ca) ...... (ca) = 0; and therefore p=0, since, by hypothesis, none of the other factors is equal to zero. Hence f(a) reduces to By hypothesis this expression vanishes for more than n values and therefore P1 = 0. of x, In a similar manner we may shew that each of the coefficients P2, P3 . P1 must be equal to zero. ...... This result may also be enunciated as follows: If a rational integral function of n dimensions vanishes for more than n values of the variable, it must vanish for every value of the variable. COR. If the function f(x) vanishes for more than n values of x, the equation f (x) = 0 has more than n roots. Hence also, if an equation of n dimensions has more than n roots it is an identity. Example. Prove that 1 (x − b) (x −c)+(x−c) (x − a) + (x-a) (x-b)=1. (a - b) (a-c) (bc) (b-a) (c-a) (c-b) This equation is of two dimensions, and it is evidently satisfied by each of the three values a, b, c; hence it is an identity. 311. If two rational integral functions of n dimensions are equal for more than n values of the variable, they are equal for every value of the variable. are equal for more than n values of x; then the expression |