SERIES. 



which values fubftituted for A, B, C, &c. will give the i. Let it be propofed to find the fum of the infinite 



transformation fought, and which will terminate by one of feries, 



thofe expreflions becoming zero, when the feries is fum- 

 mable, but when it is not the expreffion itfelf will become 

 an infinite feries, but fuch that we may give to it almoil 

 any degree of convcrgency at pleafure. 



Let us now illuitrate what has been faid by a few ex- 

 amples, remembering that the fum of a feries, whofe 

 general term is 



A + B X + C X (x - 1) + D .V (x - i) {x - 2), = 

 A A- + i B (.V + I ) .V + f C (.V + i) x{x — i) + Sec. 

 Let it be propofed to fum the feries of odd numbers, 

 ' + 3 + 5 + 7 + &c. 



Here the general term is 2 .r — i , or — i + 2 v ; fo that 

 a= — I and b =: z: whence A = — I, and B =^ 2, and 

 C = o ; whence A .->,■ + i B (.v + i ) .v = — x + x' — x 



I 



4 



+ 



+ 



'3 



+ &c 



1.4.7 4 . 7 . 10 

 where the general term is 

 I 



3--^-(3^'+ 3) (3« + 6) 

 X being fuccefTively §■, ly, z\, &c. 



Now this is of the required form, A being 



2']x{x -\- \) (x Jr 2) 



o, and 



B = — ; therefore the required fum is — 



27 



54 X (x + i) 



24 



, by taking x = \, its firft: value. If we took.v = ij, 



we fhould have the fum of all the terms of the feries, except 



the firft ; if.v::=2t, we fhould have the fum of all but the 



= .\-% which is the known expreffion for the fum of .v terms two firlt terms, and fo on : and it is by this means that we 



are enabled' to give fo great a degree of convergency in 

 thofe feries that are not fummable ; for we may affume any 

 one of the values of x, and by that means give almoft any 

 magnitude to the denominators of our converging fraftions ; 

 obferving only, that fuch of the leading terms of the feries 



of the above feries. 



Again, require the fum of the feries, 



1.2 + 2.3 + 3.4+4.5-!- &c. 



Here the general term is x (.v + i), or x' + x : 

 preceding tablet, 



by the 



* 1 / > {■ = 2 .V + .V (.V — I ). 



X = a; + .V (.V — I ) 3 ^ ^ ' 



Therefore A = o, B = 2, and C = i ; whence we have 



^ B (.r + 1 ) .V + ^ C (x + I ) .V (.r - I ) = 



{x + l) .V + 5- (k + I) A- {x - I) = 

 T (■'*■"' + 3 ^' 4" ^ •*')> ^'^ ^""^ °^ •■*•" terms, as required. 



as are not included muft be fummed by themfelves, and 

 added to the approximation found as above. As this is the 

 great charafteriitic of Stirling's method, we fliall confine 

 our futiue remarks to one or two examples, which are not 

 fummable, in order to illuftrate the nature of his approxi- 

 mations. 



Let there be propofed the feries, 



1 I I 1 



+ 



3 • 4 



+ 



+ 



r.^ -•- '"'■ ■' 



But as there is no advantage gained by the application of wh'ch is that found by lord Brounker, for the quadrature 

 this method to feries of the above kind, nor indeed to any °' '"^ hyperbola, 

 fummable feries, as thefe are commonly more readily re- Here the general term is 

 folved by feme one of the preceding methods than by this, 

 we fhall pafs immediately to feries of the fecond kind, in 

 which it poffefles a facility of application, which is perhaps 

 unattpinable by any other principle at prefent known. 



Here we mull obferve, that after the general term of any 

 feries is reduced to the form, 



+ 



B 



+ 



X {x + I) X (x + I) (« + 2) x(x + I) (.f + 2) [X + 3) 



+ &c. 



the fum of that feries is exprefied by 

 A B C 



2a; (2.-C + I) 

 taking .v = •§, li, 



I I 



4 x^ + 2 X .4 X- 

 that is, 

 I 



4 

 whence - 



4.v(.v + i) 

 &c. Now 



4 x' + 2x 



3 



i6.f+ 



3-S 



32 .v^ 



+ &c.; 



A = — , B = 



C = -^^, D 



3 



16' 



35 



32 



&c. ; 



+ 



+ 



2x(x -\- I) 3.v(.v + l) (.v+ 2) 



+ &c. 



for 



4 K -\- 2 X 



when converted into the required 



m, IS, 



+ 



3 



+ 



3-5 



j^x (x+ 1) 8 *• (.r + 1) (x + 2) 16 X (x + 1) (x + 2) (x + 3) ' 32 x{x + I) {x + 2) (x + 3) (x + 4) 



where the law of continuation is obvious, and the fum will be exprelfed by 



I • 3 • 5 



+ &e. ; 



I 1 I . « 



— + . + ? 



4x l6.v(.i-+l) 48X (x + l) (x + 2) 



+ 



128 .V {x + \) (.V + 2) (.V + 3) 



+ &c. ; 



in which the law is alfo obvious, the co-efEcients in the The original feries has, therefore, been converted into 



denominator being 4= 2% 16 = 2^ x 2, 48 = 2' x 3, another infinite feries, but with this advantage attending the 



128 = 2* X 4, &c. ; but the feries will not terminate, be- hitler, that we may give it almoft any degree of convergency 



I .... at pleafure, according to the value we give to x. If we 



4.1- (.7~+ i) ""^ u es t e ^(]■^,^J, ^, _ jjI^ which is its value in the 14th term, then 



fraftion i. ^^^ preceding feries will exhibit the fum of the original 



feries 



eaufe the original general term 



