SERIES. 



Method of fer'tes is ufed in a general fenle to denote the 

 principle upon which different authors have treated this fub- 

 jedt, as well with reference to the reverfion and interpolation 

 of feries, as to the finite and approximate fummation of 

 them. 



The doSriiie of feries is certainly one of the mod important 

 fubjefts of mathematical inveftigation, and has been very 

 appropriately denominated by James Bernoulli the Jheet- 

 anchor of analyfis ; being our only hope and lall refort, in 

 a variety of difBcult problems, which bid defiance to every 

 other method of computation. 



The fummation of feries, and the quadrature of a curvi- 

 linear ipace, are intimately connetted with each other, as 

 well in their origin as in their fubfequent progrefs. We have 

 dated under the article Quadbatuhe, that Archimedes was 

 the firft who found the area of a curvilinear fpace, which 

 he effefted by means of the fummation of an infinite feries 

 upon geometrical principles, and which is the firft inftance 

 on record of fuch an operation ; from which time, for nearly 

 two thoufand years, little or nothing was attempted relative 

 to this fubjeift ; but about the middle and the latter end of 

 the 17th century, it begun to attraft the general attention 

 of mathematicians, and has fince that time been purfued with 

 a degree of perfeverance and fuccefs commenfurate with its 

 great importance, and the general progrefs of analyfis 

 during the fame period. 



Wallis, in his Arithmetic of Infinites, feems to have 

 been the firlt amongll the moderns who drew the attention 

 of mathematicians to the doftrine of feries. Lord Brounker, 

 fir Cliriilopher Wren, Mercator, and James Gregory, alfo 

 purfued the fubjeft with confiderable fuccefs, exhibiting the 

 quadrature and reftification of difierent curves under the 

 form of infinite feries. 



In 1682, Leibnitz publifhed in the Lcipfic Aftsa memoir 

 entitled " De proportione circuli ad quadratum circum- 

 fcriptum, in numcris rationalibus," in which he gave feveral 

 numerical feries of a very novel kind, whofe fums were ex- 

 preflible in finite terms, without, however, accompanying 

 them with their demondrations ; amongll the molt curious 

 of which we may reckon the following : nih,. 



+ 



I I 



+ — + — + 

 15 24 



+ ^ + ^'^' 



+ 



I 



+ 



+ 



5 - 



+ &c. 



which is equal to the area or fpace included between the 

 curve and afymptote of an equilateral hyperbola, or \ of the 

 hyp. log. 2. 



Leibnitz alfo gave in the fame work for 1683, the fum- 

 mation of feveral other feries of a more difBcult kind, as 



The fum of an infinite number of terras of which is equal 

 to \ ; the fum of its odd terms being equal to i, and the 

 fum of its even terms equal to ^ : that is 



I -3 

 I 



+ 



( 



+ 



3 -5 



I 

 -^6 



+ 



5- 7 



+ 



7 -9 



+ 



h &c. = i, and 



6. 8 "*" 8 



10 . T2 



+ &C. 



I 



— ?• 



The fum of an infinite number of terms of the fame fe- 

 ries, omitting every three terms after the ill, the 5th, the 

 .9th, &c. as 



I 1 III 



+ 



+ r- 



I . 3 5 . 7 9 . II 13 . 15 17-19 

 is equal to the area of a circle of which the infcribed fquare 



IS I 



But if we begin at the fecond term, and thence omit every 

 three terms, as above, we fhall have 



+ 



4 6 

 Vol. XXXII. 



8 



+ 



+ 



12 14 . 16 



+ &c. 



3 6 10 15 . /2o\' 

 I 2 1 -\ ^ - — &c. =( — I 



2.10 2.10^ 2.10' Z.IO" \2I,/ 



Thefe, as we have before obferved, were not demon, 

 ftrated by Leibnitz, but this was foon alter done, and many 

 other feries invelligated, by the brothers John and James 

 BernouUi; the latter in a fmall traft " DeSeriebusInfinitis," 

 publiflied with the " Ars Conjeftandi ;" and the former in 

 vol. iv. of his " Opera Omnia." 



From the preface to the former traft we learn, that James, 

 having turned his attention to the doftrine of feries, had 

 difcovered a few which were fummable, and which he pro- 

 pofed to his brother ; who having quickly demonftrated 

 them, propofed others to James ; this led to other propofi- 

 tions, and fo on, till in a (liort time they were not only 

 able to demonftrate all Leibnitz's feries, but had difcovered 

 two general principles, which* applied with great facility 

 to a variety of new cafes ; the one of which was the 

 refolution of an infinite feries into an infinite number of 

 other feries ; and the other, the method commonly called 

 the fuinmation by fuhtraaion. We fee here that fpirit of 

 emulation and rivalry with which thefe two brothers were 

 conllantly actuated, and to which they each probably owe 

 many of their finell difcoveries. It is only to be regretted 

 that it terminated in a manner fo unworthy of their talents 

 and charafter ; particularly with regard to John, who was 

 doubtlefs at firll much indebted to his brother's inllrudlion, 

 but who, notwithftanding, indulged his refcntment againft 

 him for many years after his death, ffieking every opportu- 

 nity of afperfing his methods, and of leffening his repu- 

 tation. 



The Bernou/iis' Method of Series.— Tht principal dif- 

 ference between the methods of thefe two celebrated mathe- 

 maticians confifts in this, that James, in his " Traflatus de 

 Seriebus Infinitis," proceeds fynthetically j and John, in his 

 " Opera Omnia," analytically ; but the feries in both cafes 

 arc nearly of the fame kinds, and the fummation of them 

 depends upon the fame principles ; we fiiall, therefore, by 

 way of illuftration, abilraft one or two propofitions from 

 the former work, which will be fufficient for giving the 

 reader an idea of the fpirit of the two methods above al- 

 luded to. 



Prop. To find the fum of an infinite number of fraftions, 



whofe denominators increafe in any geometrical progreffion, 

 but whofe numerators proceed accordmg to the natural num- 

 bers, or polygonal or figurate numbers, of any denomi- 

 nation. 



(jafe 1. — When the numerators proceed accordmg to the 

 natural numbers, that is, when they form an arithmetical 

 progreffion. 



Let the propofed feries, whofe fum is required, be 



a a -\- c a 



Nn 



+--±-li+.=. 



This 



