to the coefficient of the next following term. If the series be expressed by A+B+C+D+E+F+G+H+, &c., then That is, each term after the second is equal to the one next preceding, multiplied by x, plus the second next preceding, multiplied by x2; hence, all the terms after the first two recur according to a definite law. ART. 340. The particular expression by means of which any term of the series may be found when the preceding terms are known, is called the scale of the series, and that by means of which the coëfficients may be found, the scale of the coefficients. Recurring series are said to be of the first order, second order, &c., according to the number of terms contained in the scale. Thus where each term after the first is equal to the preceding, multi b plied by x. In this case b x is termed the scale of the b series, the scale of the coëfficients, and the series is said to α be of the first order. This is the most simple form of a recurring series. ART. 341. To find the scale of a series. When the series is of the first order, the scale is easily determined, being the ratio of any two consecutive terms. When the series is of the second order, the law of the series depends on two terms, and the scale consists of two parts. Let p+q represent the scale of the recurring series A+B+C+D+E+F+, &c. Then the 3rd term the 4th term the 5th term C=Bpx+Aqx2; . D=Cpx+Bqx2; E=Dpx+Cqx2; &c. The values of p and q may be found from any two of these equations. Taking the last two, and making x=1, since the scale of the series is the same, whatever be the value of x, we have Since these formulæ were obtained by supposing x=1, therefore, in substituting the values of B, C, D, &c., x must be considered 1. Ex. Find the scale of the series 1+2x+3x+4x3+5x+, &c. Here A=1, B=2x, C=3x2, D=4x3, E=5x1, &c. Making x=1, and substituting the values of B, C, D, &c., Thus, the 4 term, 4x3=2פ×3x2+2x×−1×x2. Other exercises will be had in finding the sums of recurring series. ART. 342. In a recurring series of the third order the law of the series depends on three terms. If we let p+q+r represent Making x=1, the values of p, q and r, are readily found, (Art. 158); and in a similar manner the scale may be determined in the higher orders of recurring series. In finding the scale of a series we may first make trial of two terms. If the results thus obtained do not reproduce the series we may try three terms, four terms, and so on, till a correct result is obtained. If in any case we assume too many terms, the redundant terms will be found equal to zero. ART. 343. To find the sum of an infinite recurring series whose scale of relation is known. Let A+B+C+D+E+, &c., be a recurring series whose scale of relation is p+q; then If the series be continued to infinity, the last term may be considered zero. Then if S represent the required sum, by adding together the corresponding members of the preceding equalities, and observing that B+C+D+, &c., =S—A, we have __A+B—Apx, which is the formula for 1-px finding the sum of an infinite recurring series of the first order. In a manner similar to the preceding, the sum may be found when the scale of the series consists of three, four, &c., terms. REMARK. Every summable infinite series, of which recurring series are only a particular class, may be supposed to arise from the development of a rational fraction; hence, to find the sum of an infinite recurring series, is to find the generating fraction of the series. 1. Find the sum of the infinite recurring series 1+3x+5x2 +7x+9x+11x+, &c. Here A=1, B=3x, C=5x2, D=7x3, E-9x^, &c. Making x=1, and substituting in the formula (Art. 341), we In each of the following series find the scale of relation, and the sum (S) of an infinite number of terms. 2. 1+6x+12x2+48x3+120x1+, &c. 7. 1+2x+8x2+28x+100x+356x+, &c. Ans. p=3, q=2; S=; 1-x 1-3x-2x2 8. 1+3x+5x2+7x3+9x1+, &c. Ans. p=2, q=—1, S=; 1-2x+x2 1+x 9. 12+22x+32x2+42x3+52x2+62x2+, &c. ART. 344. To revert a series is to express the value of the unknown quantity in it by means of another series involving the powers of some other quantity. Let x and y represent two indeterminate quantities, and let the value of y be expressed by a series involving the powers of x; thus, y=ax+bx2+cx3+dx1+, &c., (1). in which a, b, c, d, &c., are known quantities; then to revert this series is to express the value of x in a series containing the known quantities a, b, c, d, &c., and the powers of y. To revert this series, assume x=Ay+By+Cy3+Dy', &c. (2), in which the coëfficients A, B, C. . . . are undetermined. Find the values of y2, y3, y1 . . . from (1), thus, y2=a2x2+2abx3+(b2+2ac)x1+ Substituting these values in (2), and arranging, we have 0=Aax+Abx2+ Acx3+ Ad\x2+, &c. and since this is universally true, whatever be the value of x, the coefficients of x, x2, x3, &c., will each =0. (Art. 314, Cor.) Hence, we have Ad+Bd2+2Bac+3Cab+Da1=0, ... D=_a2d-5abc5b3 απ ART. 345. If the given series has a constant term prefixed, thus, y=a+ax+bx2+cx3+dx1+ assume y-a-z, and we have z=ax+bx2+cx3+dx1+, &c. But this is the same as (1) in the preceding article, except that z stands in the place of y; hence, if z be substituted for y in [(3), Art. 344], the result will be the required development of x; and then y-a' being substituted for z, the result is |