SERIES. 



Now the funflion/(A:) is given, being the general term 

 of the feries and the objeft of enquiry is, from this given 

 fundlion to determine the two funAions p (x) and (^ [X — I), 

 of which it is the difference ; for the former of thefe, i? (x), 

 will then be the fum of the feries to the term / inclufive. 



To iUiiltrate this by, a familiar example ; let the propofed 

 feries be 



1+3 + 5 + 7 -I 9' &c. 



the o-eneral term of which is 2 .r — i ; therefore, 



Q («) - : (a:- 1) = 2.V- i; 



whence we have obvioufly <p (x) == x\ for 



,v- — (x — i)' = 2.V — 1 ; 



therefore .v" is the fum of x terms of the above feries. 



Again, let it be required to find the fum of x terms of 

 the (cries 



I, 7. 19' 37' &e- 



of which the general term 



/(.r) = 3 .V--3 .v+ I, 

 therefore, 



(?.«-? (.Y - I ) ^ 3 .1- — 3 A- + I ; 



confequently ::■ {x) = .r , the fum of x terms of the feries. 

 In thefe two examples, the finding the fum C (.v) from 

 the diiTerence. is extremely fimple ; but in the generality 

 of cafes ir is far from being fo obvious, and even in fome, 

 it is inipoffible to exhibit the fum of the feries in any other 

 manner tiian by another feries ; but as in the latter cafe 

 the transformed feries may be made to poffefs almoit any 

 deeree of convergency we pleafe, this method of fumma- 

 tioii is ftill attended with important advantages, and even 

 more perhaps in the latter cafe, than in any other ; becaufe 

 moll, if not all, fummable feries, may be fummed on fome 

 other principles ; viz. either by the method of recurring ferks, 

 or by the differential me/hod, or by increments; wliile the 

 transformation of a flowly converging feries, into another 

 of rapid convergency, is frequently extremely difficult to 

 effeft on any other principle than that of Stirling's, or 

 fome other tantamount to it. 



From vvliat is Hated above it appears, that the principal 

 objeft of enquiry is, in what manner we are to determine 

 a funftion from the difference between two ftates of it be- 

 ing given. In the examples we have chofen for illuftration, 

 the funftion whence the difference is derived is extremely 

 obvious ; but this in many cafes is, as we have before ob- 

 ferved, attended with fome difficulty. In this refpeft the 

 fummation of feries refembles in a great degree the inverfe 

 method of fluxions. There is little or no difficulty in any 

 cafe in fir.ding the fluxions of any propofed quantity ; but 

 the finding of a fluent of any given fluxion i« far from pof- 

 feffing the fame facility. So alfo in the prefent cafe, if the 

 queftion was to find the difference between two different 

 ftates of a given function, we ftiould find the operation di- 

 rect and fimple ; but the converfe, or the finding the func- 

 tion from the difference being known, is indirect, and fre- 

 quently difficult to be determined. 



It is obvious alfo, that two different funftions, which 

 differ from each other only by fome conftant quantity, vi-ill 

 give the fame difference, and, confequently, a given differ- 

 ence may give rife to different funtlions, the fame as happen 

 in finding fluents, and it will therefore be neceffary in this 

 cafe, as in that, to have recourfe to a correSion, which will 

 be found in the fame manner as is praftifed in that calculus, 



2 



•viz. by finding the value of the feries, when the variable 

 quantity is made equal to zero, or fome determined magni- 

 tude. 



0/ the general Term of a Series With regard to the gene- 

 ral term of a' feries, it is difficult, particularly within our 

 limits, to lay down any fixed or conilant rule for its deter, 

 mination ; it is befides feldom neceflary, as the law of the 

 feries is commonly prefented in the terms of tie feries Itfclf; 

 we fliall leave this determination, therefore, as in faft it 

 muft be in mofl; cafes, to the ingenuity of the analylt, and 

 fliall proceed immediately to the other fubjcfts of inveitiga- 

 tion. 



It may not, however, be amifs to ftate ; that in fuch 

 feries as have any order of their differences vanifli, the gene- 

 ral term is always of the form 



An'" + Bn"'^' + Cn"'-'' -|- Dm"-^ + Sec. 



where m denotes the order of the differences that vanifli, 

 and .<j the number of terms from the beginning. The 

 values of A, B, C, D, &c. being found by making n fuc- 

 ceffively equal to I, 2, 3, &c., and equating the refults 

 with the lit, 2d, 3d, &c. terms of the feries. 



Of the Transformation of a given FunSion to an equivalent 

 one.of a different Form. — Since we fliall confine ourinvelliga- 

 tion only to thole feries whofe terms are either integers or 

 rational fractions, it is obvious that the general term mult 

 alio be fome rational funftion either of the form, 



a -\- bx + ex' -j- </.v' -f- &c. or 



a + bx 'r ex' -f dx' + &c. 

 a' + b'x + c'x + d'x' + &c. 



and our object is to transform eithrr of thofe general forms 

 into others, whence the general fui:ftion from which they 

 have been derived may be the more readily determined. 

 Different transformations may be emphiyed for this purpofe ; 

 but the moff general, and that, in fact, to which Stirling 

 principally confines himfelf, is to transform the above gene- 

 ral terms into other equivalent ones of the form 



A -f B .V -f- C .r ( .v — 1 ) + D .1- {x — l)(x — 2) + Sec. or 



A B C ______ 



.7(J+ 1) "^ X {x + l){x + 2) '^ X (x + 1) {x + 2) {x + z) 



from either of which the general funftion whence they have 

 been derived may be readily determined. For it is obvious 

 that the firft is equal to the difference between the two 

 fimilar funftions 



A X + i B (x + I) X + 4 C (x + i) X {x - i) 

 + ^D {x + l) X {x — 1) {x — 2) + Sec. 



and 



A (;; - I) + 4 B .r (.V - I ) -I- 4 C x (x - 1) (x - 2) ' 

 + iH X (x - 1) (x - z) {x - z) + &c. 



For by fubtracting thefe one from the other, we have 



A + B X + C X (x — i) + !> X (x — i) {x - 2) + Sic. 



And therefore, from what has been Rated, the firft of the 

 above formulas will be the general fum of that leries of 

 .which the general term is 



A + B.-« + Cx{x- l) + Dx(x- I) {x-2) + &c. 



And in a fimilar manner it may be Ihewn, that the fecond 

 general form is equal to the difference between the two 

 fimilar fundtions 



