SERIES. 



Again, let N reprefent the nth term, and S the fum of 



N 



a + 



S = na -\- 



I 

 « (h- l) 



D' 



+ 



(„- I) (n- 2) 



D' + 



n (n 



I . 2 



I) {n-2) 



n terms ; then will 



("- i) ("- 2) ("-3) 

 1.2.3 



" ("— i) (" - 2) (« — 3) 



D" + 



which latter expreffion for the fum is, as we have above 

 obferved, due tc M. Montinort. The fame author alfo 

 publifhed in the Pliilofophical Tranfaftions for 1718, fome 

 other formulae for the fumm-tion of feries ; but as thefe are 

 nothing more than particular cafes of the method of incre- 

 ments, w ■ fliall not notice th m in this place ; but refer the 

 reader to ihe article Increments, for an illuftration of the 

 method of fummation as depending upon thofe principles, 

 firft pubh(hi 1 by Dr. Brooke Taylor, in his." Methodus 

 Inc'einentorum," 1715. 



4. L- Moivre's Method of Series. The next author who 

 made any confiderable inrip'-ovement in this theory was De 

 Moivre, to whom we owe ti.j doftrineof Recuuring Series, 

 on the princ'ples of which we have fpoken at fome length 

 under that a; cle of the prefent work ; we fliall not there- 

 fore enter agau. upon the fubjeft in this place, but confine 

 ourfelves to an illuftration of his method for finding fum- 

 mable feries, which is not referred to in the article above 

 mentioned ; it was firft given by liim in his " Mifcellanea 

 Analytica," I??©. 



Let there be affumed any feries, and let this be multiplied 

 by any binomial or trinomial faftor, fuch that the refulting 

 feries (hall have its powers of x recurring again in the fame 

 order ; then, by equating the refulting feries to o, and tranf- 

 pofing the negative terms, a new numerical feries will arife, 

 the (urn of which will be given. 



Thus, let there be taken the feries 



I + ix + ix 

 Multiplying this by x 



I I 



- I + 



X* + &c. = S. 



- X + — 

 2 2 



Whence, making x 



+ 



(x - 1) S. 



&c. = I. 



3 3-4 

 Again, aflume 



1 + Ix + 5-.r' + ix' + &c. 

 Multiplying by x' — i, we have 



2 



1 . 2 



- I x i A-' 



2 1.3 



+ 



2 . 4 



= (.v'-l) S 

 where making again .r = i, we have 



3-5 



+ Sec. 



m 



2 2 



+ occ. = -^, 



2 



1,1 I I 



+ .— + - — r + 



I 3 



, r h &C. = -^. 



'•3 2-4 3-5 4-f> 5-7 4 

 As another example, let the fame feries 



I + 



+ >* + ix' + ix' + &c. = S 



be multiplied by 



{2X- I) (3*- 1) =6x^- sx + I, 



D'"-h &c. 



D" + 



and we have 

 9 



X + 



I . 2 

 = (2.r 



where, by making .v 

 lowing feries : 



23 



•3-4 



23 

 1.2.3 2 



- I) (3-^--0S 



D'" 



&c. 



«■ + 



38 



3 -4 



.•c^ + 



57 



3-4- J 



and X = J, we have the two fol- 



1.2. 



23 



- + 



38 



3 4 



+ 



•3- 



38 



8 + 



3-4 



— +- 



27 3 



i7_ 

 .4. 



57 



4.5 



I 



I 



81 



&c. = -' 



+ &c. = 



1.2.3 9 



The law of both which feries is obvious, the numerators 

 being in arithmetical progreffion. This method is not much 

 different in principle from the fecond method of Bernoulli 

 above explained. 



5. Stirling's Method of Series. In the recurring feries of 

 De Moivre, each term is connefted with a certain number of 

 the preceding terms, by a conttant and invariable law, but in 

 the feries confidered by Stirling, in his " Methodus Differen- 

 tialis," 1730, each term is a certain function of the number of 

 terms from the beginning, or from fome determinate term of 

 the feries ; which funftion may therefore be confidered as 

 the general term, and the method of fummation depends on 

 the following principles. 



Having firlt determined the general term of the feries in 

 fome funftion of x, its dillance from the beginning, or 

 fome determinate term of the feries ; it follows, tliat the 

 fum of all the terms to that place will alfo be fome fundion 

 of X. Therefore, if .r' is made to denote the diftance of any 

 other term from the fame point, the fum to that term will 

 be the fame funftion of x', as the other fum is of .v ; and 

 each term of the feries may be confidered to reprefent the 

 difference between two confecutive fums, or the difference 

 between two fimilar funftions, viz. of .v — i and .v ; and 

 the objeifl; of the author is to determine what thofe fums or 

 fundtions are from the difference between them being given. 



To bc' a little more explicit, if there be any feries of 

 quantities 



a, b, c, (I, . . . . t°, /, /', &c. 



proceeding from the firft a, by any imiform law, either in- 

 creafing or dccrcafing ; and if x be taken to reprefent the 

 diftance of any term, as /, from the beginning of the feries, 

 or from any term in the fame, then will / be exprtffiblc by 

 fome funftion of x ; t' by the fame funftion of .r + i j 

 t" by the fame funftion of x -)- z, &c. : denoting therefore 

 this funftion by / {x), we fliall have 



P=f{x-i),t =/ (.v), t' =f (x + I), &c. 



Alfo if /S /,/',/", &c. denote the fums of all the 

 terms from the beginning to the terms t°, I, /', /", &c. 

 refpeftively, thefc Icvcral fums will alfo be fome funflion 

 of X — I, X, .V -f- I, jr + 2, &c. which we may denote by 



r = 9 {x-i),f=<p (.v),/'=<f ix+ 1): 



whence we draw immediately 



/-/" = /, or 9 (.v) -. p (.r - I ) =/ (.. ). 



Now 



