ON RECURRING SERIES. (179.) A recurring series is one, each of whose terms, after a certain number, bears a uniform relation to the same number of those which immediately precede. (180.) It is obvious that a variety of infinite series will arise from developing different fractional expressions: those however which generate recurring series are always of a particular form. (181.) The fraction a a' + b'x' for instance, is of this kind, for the series which arises from the actual division is recurring thus: where it is obvious that each term, commencing at the second, is equal to that which immediately precedes multiplied by quantity is called the scale of relation of the terms, or b'x which of relation of the coefficients; therefore, representing the terms of the series by A, B, C, D, &c., we have here we may observe, that the coefficients of A, B; of B, C ; of C, D, &c., are the terms of the denominator of the generating fraction taken in reverse order. (182.) The fraction a + bx is another of this kind; for if this be developed as that above, and similar substitutions be made, there will be found to result where each term, commencing at the third, is equal to the two im mediately preceding multiplied respectively by c'x2 b'x which a' is therefore the scale of relation of the terms; also, the coefficients of A, B, C ; of B, C, D, &c., are the terms of the generating fraction taken in reverse order. (183.) The fraction a+bx+cx2 is also one of the same kind, as its development will show; the scale of relation of the terms, the fourth term. And, in general, the development of any rational fraction of the form will be a recurring series, in which any term, commencing at the m + 2th, will be equal q'xm+1 α p'xm to the m + 1 preceding multiplied by c'x2 b'x ܕܙ܂ respectively, which is there - c'x2, — b'x, being the several terms of the denominator taken in reverse order, the first term 1 being omitted. PROBLEM I. To find the sum of an infinite recurring series. Let A+B+C+D+......+K+L+M+ N represent a recurring series, and let it be supposed such, that each term, commencing at the fourth, depends upon the three preceding; then, as in Art. (182), we shall have, by supposing the terms in the generating fraction to be p, q, r, s, the following equations, viz. SK + rL +qM + PN = 0, and taking the sum of these equations, we have which, by putting s for the sum, becomes the same as 8(S-L-M-N) + r (s—A—M-N) + q (s-A-B-N) +P(s-A-B—c) = 0; from which equation we get s = P(A+B+C)+Q(A+B+N) + r (A + M + N) +8 (L+M+N); p + q + r + s so that the sum may be determined from having the three first, and three last terms, with the scale of relation given; but if the series be infinite, and decreasing, the three last terms will vanish, and the sum will be P(A+B+C)+(A+B)+rA _ ▲ (p+q+r)+B (p+q)+cp = 1. Required the sum of the infinite recurring series 1+2x+8x2 + 28x3 +100x1 + 356x + &c. Here the scale of relation is 2x2, 3x. .. the third term, c=2x2A + 3xв, whence consequently, 8 = — 2x2, r = -3x, q= 1, and p=0. 8=— 2. Required the sum of the infinite recurring series 1 + 2x + 3x2 + 5x3 + 8x1 + &e. the scale of relation being 2, x. 1 + x Ans. 1 3. Required the sum of the infinite recurring series · 1 + 3x + 5x2 + 7x3, &c., the scale of relation being — x2, 2x. 4. Required the sum of the infinite recurring series 3+5x+7x2 + 13x3 + 23x1 + &c. the scale of relation being — 2x3, x2, 2x. PROBLEM II. To find the sum of any number of terms of a recurring series. This may be effected by means of the expression for s in the preceding problem, but more conveniently by subtracting from the sum of the series continued to infinity, the sum of all those terms which follow the nth; thus, if the nth term of the recurring series A+B+C+ &c. be T, then, putting s for the sum of all the terms to infinity, and s' for the sum of those to infinity which follow T, we shall have, by last problem, s -s' = = A(p+q+r)+B(p+q)+cp-v(p+q+r)—v (p+q)-wp p+q+r+s (A—U)(p+q+r)+(B—v)(p+q)+(c—w)p p+q+r+s + .. C = x2A + 2xв, whence A 2xB + C = = = 0, ... 8 = x2, r =—2x, q 1, p=0, also u= (n + 1)x"; *The finding the sum of a finite number of terms of a recurring series supposes that the general term of the series is previously known: to discover the general term is, however, by far the most perplexing part of the problem, it being often attended with considerable difficulties. The only general way in which it can be discovered is derived from considering the generating fraction which may be expanded, and the general term of the resulting series ob tained, by the MULTINOMIAL THEOREM. |