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) 1-px - qa2 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=29+(a, – pa,).X + (ag-pa, -qan)x2. 1-pc-q22 1-px-2x2---23 491. From the result of Art. 489, we obtain ao +(a, – pa.). =a, +972 +222+.. +an-13n+1 + ; 1-px-qxo from which we see that although the generating function a,+(a, -pa) 1-px-922 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 do +aqx+x2 + ... only if the remainder (pan-1+90,-) .x"+qan-12+1 1-px — 932 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-q3c2; then -1+7p-q=, –43+p+79=0; whence p=1, q=6; and the scale of relation is 1-2-622. S=1-7X – 22 – 43x3 – - 6xPS= -6x2 +42x3+ 1-83 S= 1-2-6x23 which is the generating function. 1-88 If we separate into partial fractions, we obtain 1-X-6x2 2 1 1+2x 1-32 By actual division, or by the Binomial Theorem, 2 =2[1-2x + (2.c)2 - ... +(-1) (2.c)'] -[1+3x+ (3x)2 + ... + (3x)"]. [2(2)(-1)" — 3"]xr={(-1)" 2r+1 — 3"}ær. EXAMPLES XLVI. a. Find the generating functions of the following series : ...... 8. 1-3x+5x2 — 723 +934 – 11.05 + Find the generating function and the general term in each of the following series : 9. 1+5x+9x2 + 13x3 + 10. 2—3 +522 — 72c3+.. 11. 2+3x+5x2 +933 + 12. 7–6x+9x2 +274 + ...... 13. 3+6x+14x2+36x3 +98x4 +276x5 + SUMMATION BY THE METHOD OF DIFFERENCES. 493. Let un denote some rational integral function of n, and let U7, U2, U3, U4, ... 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 Un when a certain number of the terms un, U2, U3, U 42 are given. From the series U7, U2, U3, U42, U 59 obtain a second series by subtracting each term from the term which immediately follows it. The series U2 — Uy Uz — U2, U4 – U3, Ug — U 49 ... thus found is called the series of the first order of differences, and may be conveniently denoted by Au Auz Aug, Auq, ... By subtracting each term of this series from the term that immediately follows it, we have Aug-Au Aug-Aug Auf-Aug ... which may be called the series of the second order of differences, and denoted by A,U1, A2U2, A3u3, ... 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 AgUpg A Un Ag2ry ... respectively. From the law of formation of the series 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 u, =u, + Δυ, and Δυ, =Δυ, + Δρυς. Uz=wy +2A4, +42Uz. In an exactly similar manner by using the second, third, and fourth series in place of the first, second, and third, we obtain Aug=Au; +24,4, +Azuz. Un=u, +3A4, +34,4,+Azuj: 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. For sup pose that n(n-1) Un+1=4, +nAu,+ A,+ .+nC Arun + + Anuzi 1.2 then by using the second to the (1+2)th series in the place of the first to the (n+1)th series we have n1) AUm+1=Au, +n4,4, + , Cr14, 1.2 t. + Antilly By addition, since Un+1+Aun+1=Un+2, we obtain Un+2=u, +(n+1)Au, + ... +("C+"C,-1)4,47 + +Antily -r+1 But +1)x C-1= =n+1 Cr 1.2.3 ... (r-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 U4) therefore it holds for Ug and therefore universally. Hence (n-1)(n-2) Un=4, +(n-1)Au, + + An-1U 1.2 n+1x»C-1 494. This formula may be expressed in a slightly different form, as follows: if a is the first term of a given series, dį, da, dg, ... the first terms of the successive orders of differences, any term of the given series is obtained from the formula (n-1)(n-2) an=a+(n-1)d, + d,+ 12 13 (n-1)(n-2)(n-3) dgt .... 495. To find the sum of n terms of the series U12 U2, U3, U42 in terms of the differences of Un Suppose the series U7, U2, U3, is the first order of differences of the series V1 V2 V3 V42 ..., then vn+1=(Un+1- vn) +(un— Vn-1) + ... +(v2— v.)+v, identically; :: Pn+1=Un + Un-it... tuz+u, +v1 Hence in the series 0, V2 V3, V40 V5 the law of formation is the same as in the preceding article; n(n-1) 1.2 that is, uz +U2 + Uz + tun n(n-1) n(n-1)(n-2) 13 -A,U, + ... + Anug 496. If, as in the preceding article, a is the first term of a given series, dı, də, dz, the first terms of the successive orders of differences, the sum of n terms of the given series is obtained from the formula n(n-1) n(n-1)(n-2) Sn=na + -d,+ 2 13 d, + 3 |