SERIES. 



be expreffed by the fluxional formula 



where, becaufe it is given in finite terms, A will always 

 likewife be had in iinite terms, and confequently, alfo, the 

 value of B. And in the fame manner as we have 



ax' + b a' + " + c x' + ''" + &c. = A 



r<j;rf + (r + «) bx'^" 4- (r + 2 «)*•'■+=" + &c. = 



.V A 

 {r-/)A+— -B; 



fo alfo 



riax + (r-f n) [s -\- n) b x' -^ " +(r+2n) {s + 2 n) c x^-^ '" 

 + &c. = 



(j — 6) B + - — = C &c. &c. 



X 



The three preceding cafes are the firft, fecond, and third 

 propofitions in tlie author's chapter on feries, which contains 

 four other propofitions equally general and important ; but 

 for thefe we mult refer the reader to the tradt itfelf. 



Since the publication of Simplon's work above referred to, 

 a variety of other treatifes have appeared either wholly or in 

 part devoted to this fubjeft, betides numerous memoirs in all 

 the principal academies and learned focieties in Europe. It 

 will be impoflible to enter upon thefe at any confiderable 

 length within the limits of this article, and we (hall there- 

 fore merely felcdl two or three of the principal authors 

 whofe methods are the moft eligible for the purpofes of ge- 

 neral fummation. 



Euler, in this, as in every other branch of analyfis, has 

 diftinguiihed himfelf by the many new lights he has thrown 

 upon this theory, and the general and elegant inveftigations 

 that he has given of many very interefting problems relating 

 to this doftrine. Thefe inveftigations are found in various 

 memoirs in the Afta Petrop. and in his " Inllitutiones calculi 

 differentialis," as alfo in the firll volume of his " Introduc- 

 tio in Analyfin Infinitorum ;" many of thefe, however, may 

 be referred to the Mi-z/jo^ 0/" Increments and Recurring 

 Series, which have been already treated of under thofe ar- 

 ticles ; the theory of circular feries is alio handled in his 

 ufually mafterly manner, but for our purpofe we (hall pre- 

 fer adopting the method employed by Landen in his " Ma- 

 thematical Lucubrations," and (hall therefore, in this place, 

 limit our obfervations to Euler's differential method. 



Euler's Differential Method of Series. — Let there be pro- 

 pofed the general feries 



S — a X + b x' + c s5 -I- (/ .f' -I- &c. 



in which a, b, c, d, &c. are conftant and pofitive quantities, 

 .-c being indeterminate. This feries Euler transforms into the 

 following equivalent feries, ■y/'z. 



a -H 



+ 



A J -f 



(I - .r) 



- A3 (7 -I- &C, 



(I - x) 



A' a 



'n which A a, A^ a, A^ a, denote the firft terms of the 

 firft, fecond, third, &c. differences of a, b, c, &c. obferving 

 that the leading term is always fuppofcd to be taken from 

 the following, fo that when the terms diminifli, this dif- 

 ference will hi negative. It is obvious here, as in the ether 

 differential feries we have had occafion to notice, that when 



any order of differences vanilh, the transformed feries will be 

 finite, but in other cafes infinite, the fame as that whence it 

 is derived. 



Let, for example, the feries 



S = A- -h 2 .v" -(- 3 .V* + 4i* -I- &c. 



be the one propofed. Here the firft differences are 1, i, i, 

 &c. and, confequently, the fecond differences are zero; that 

 it, we have a = i , and A a ^ 1 ; fo that we have 



S = a -)- 



1 — .V 



A a 



(I-.V)- (l-.r)' 



Hence, by fubftituting .V = i, |, \, &c. we obtain 



.i;i= i;S:=I-f 2 -t-3 + 4+&C. 



= 2 



i^-W 



Again, let the propofed feries be 



S = .1- -I- 3 .r* + 5 aJ -}- 7 a' -(- &c. 

 Here a ■=. 1, A a = 2, A" a = o ; therefore 



x' -^ X 



a -I- 



;Aa = 



I — X { 1 — *)* ~ ( 1 _ ;f )' 



Making therefore, as before, x = i, \, J-, &c. we have 



1 + I 



= 3 



.r = I ; S = I -i- 3 -t- 5 + 7 + &c. = 



2 2 4 8 16 



I T ^ C 7 



* = -;S = --i--i-f-^-f-^-f- &c. = , ^_. 



3 3 9 27 81 (i-O' 



Without farther examples, it is obvious, that a moft ex- 

 tenfive clafs of fiimmable feries may be drawn from this one 

 fimple principle, by merely changing the values of x ; and 

 thofe of a, b, c, d, &c. being fo aflumed, that a certain or- 

 der of their difference may vanifh, which will always hap- 

 pen, if they be made to reprefent any order of polygonal or 

 figurate numbers, or any order of powers whatever. This 

 method, however, is not limited to finding fummable feries, 

 it may frequently be employed to great advantage in ap- 

 proximating towards the real value of (lowly converging fe- 

 ries that are not fummable in any finite form, as for exam- 

 ple, the feries 



I ' I I o , , 



I 1 1 V &c. = hyp. log. 2 ; 



putting this under the form 



i 



* + 



+ 



we have a = I, A a = ~ -5, 



3 



A a 4- 



-I- &c. 



A3 a = - ^ &c. 



AV -H «cc. 

 wiU 



{i-x) 



