If we develop this fraction in ascending powers of x as explained in Art. 457, we shall obtain as many terms of the original series as we please; for this reason the expression a+(a-pa)x * is called the generating function of the series. The summation of the series is the finding of this generating function. If the series is of the third order, S= Sa+(apa)x (a,-pa,-qa)x2 491. From the result of Art. 489, we obtain may be used to obtain as many terms of the series as we please, it can be regarded as the true equivalent to the infinite series vanishes when n is indefinitely increased; in other words only when the series is convergent. 492. When the generating function can be expressed as a group of partial fractions the general term of a recurring series may be easily found. Example. Find the generating function, and the general term, of the recurring series 1-7x-x2-43x3 *Sometimes called the generating fraction. Let the scale of relation be 1-px-qx2; then -1+7p-q=0, -43+p+7q=0; whence p=1, q=6; and the scale of relation is By actual division, or by the Binomial Theorem, 2 =2[1−2x+(2x)2— ... +(−1) ̃(2x)*] 1+2x Whence the (r+1)th, or general term, is [2(2′′)(−1)”—3′′]xr={(−1)r 2r+1—3r}xr. 8. 1-3x+5x2- 7x3+9x1-11x5+ Find the generating function and the general term in each of the following series: SUMMATION BY THE METHOD OF DIFFERENCES. 493. Let un denote some rational integral function of n, and let u1, u, ug, u,... denote the values of un when for n the values 1, 2, 3, 4, are written successively. ... We proceed to investigate a method of finding u, when a certain number of the terms u1, U2, 3, 49 are given. ... ... From the series u1, u, ug, U49 U59 · obtain a second series by subtracting each term from the term which immediately follows it. The series U2-U19 U3-U29 U4 — U39 Uz — U49 ••• thus found is called the series of the first order of differences, and may be conveniently denoted by Au1, Aug, Aug, ▲u4, ... By subtracting each term of this series from the term that immediately follows it, we have From this series we may proceed to form the series of the third, fourth, fifth, orders of differences, the general terms of these series being Agur, Aur, Aur,... respectively. ... it appears that any term in any series is equal to the term immediately preceding it added to the term below it on the left. Thus uu+Au, and Aug=Au2+Au1. By addition, since u2+Au2=ug, we have ug=u+24u+ Δ21. In an exactly similar manner by using the second, third, and fourth series in place of the first, second, and third, we obtain Aug=Au2+2Aqu2+Ağu2· By addition, since u2+▲u ̧=u, we have U1=u2+3Aμ2+3A1⁄2μï+A ̧μ¿• So far as we have proceeded, the numerical coefficients follow the same law as those of the Binomial Theorem. We shall now prove by induction that this will always be the case. pose that For sup then by using the second to the (n+2)th series in the place of the first to the (n+1)th series we have By addition, since un+1+▲un+1=Un+2, we obtain Un+2=U2+(n+1)Au1+ +("Cr+"Cr−1)▲,u1+ +An+11⁄41• But ... ... (n+1)n(n−1)... (n+1−r+1) -1)r Hence if the law of formation holds for un+1 it also holds for Un+2, but it is true in the case of u, therefore it holds for u, and therefore universally. Hence Un=U1+(n−1)Au1+ (n-1)(n-2) ... Али + +An-141• 494. This formula may be expressed in a slightly different form, as follows: if a is the first term of a given series, dy, d2, dg,... the first terms of the successive orders of differences, any term of the given series is obtained from the formula an=a+(n−1)d2+ (n-1)(n-2). (n − 1) (n − 2) (n − 3) 12 -d2+ B 495. To find the sum of n terms of the series d2+.... Suppose the series u1, U2, Ug, is the first order of differences of the series ... - then vn+1=(n+1-Un) + (Un− Un−1) + ... + (v1⁄2−v1) + v1 identically; •• Un+1=Un + Un-1+ • + u2+U1+ v1• Hence in the series 0, V2 Vz V4 V5 U19 Ида Иза U4 Au, Aug, Aug...... the law of formation is the same as in the preceding article; 496. If, as in the preceding article, a is the first term of a given series, d1, d2, dз, · the first terms of the successive orders of differences, the sum of n terms of the given series is obtained from the formula |