SERIES. 



This is obvioufly equal to 



ad 



a a a '^ _l o, 



T^Vd'^bd^^Td^^^''-'" b'd-b 



c c c_ __ c d 



^U^bd'^bd>^id-i 



In a fnnilar manner is found the fum of the t'eries, when 

 the numerators are fquares, cubes, &c. from which the 

 author draws the following refults ; viz.. 



c <^ o, _ ' ^ 



'^ n^'^ hl^'^ bd' - bd 



bd^ 



bdi 



+ &c. 



+ occ. 



bd'-bd' 



: &C. 



Each of which feries being geometrical, are found by the 

 known rules for fuch progrefiions ; and it is obvious that all 

 thefe fnms, except the fiilt, are alfo in geometrical progref- 

 fion ; the fum of which, vix. of 



c d c d c d c d 



Fd^^b ^ b~d^~^Jd "^ VeP^TJ' '^ blF^b d' 



Nat. Num. 

 Trian. Num. 

 Fig. I ft order 

 Squares 

 Cubes 



1234 



— + -^ + -. + -.+ Sic. =z 2 

 2 2^ 2' 2' 



1 ? 6 10 . 



2 2 2' 2* 



-^ + ^ 

 2 2- 



-- + — f- &c. = 8 



2 2^ 2' 2* 



8 27 64 . 



2' 2' 2* 



+ Sec. = 

 to wliich therefore adding 



d 



- b{d- i)' 



a d 



b{d- if ' 



have 



ad 



As an illuftration of the fecond method, that is, of 

 fummation by fubtraftion, we (hall give an abftraft of James 

 Bernoulli's fifteenth propofition, which is as follows. 



Prop. — To find the fum of an infinite feries of fraclions, 

 whofe numerators conftitute a feries of equal numbers, and 

 denominators, a feries of triangular numbers, or of their 

 multiples. 



b(d — I) From the feries 



a a 



— + — 

 c 2 c 



+ 



— for the fum of the propofed feries. 



a 

 3-^ 



a a 



— + — 

 4<: SC 



Hd-i)' f^^btraft 



Cafe 2. — When the numerators of the fraftions proceed 

 according to the triangular numbers. Let 



6 6- 10^ , we have 



.- + T-rr + &c. 



a a a a a „ a 



— + — + — + - + >- = s 



2 C 3^ 4f ^C Of C 



?,<: 



+ 



a a 



2 c be 



a a a 



* ^.bd bd- ' bdi 



30 c c 



be the propofed feries. This may be refolved as follows : the double of which = 1 ■ + 2~ "i 1" — — 



f^ '^ ' c 2 c OC IOC ICC c 



12c 

 a 



20c 

 a 



c c c c c a 



1 ^ b'd^ Vd'-^ TJ^ '^ rr^b 



ZC 2c 2 C e, __ ^ '^ 



^bd+bd^^ rd^ ^ ^"^ ~ TT-i, 



^ bd'^ bdi^ bd'-bd 



which laft is a feries of fraftions of the form propofed, their 

 denominators forming the feries of triangular numbers, mul- 

 tiplied bv the conftant quantity c. Thus in numbers ; if 

 from the ieries 



' I.I I 



- +-+-+- + &c. 



2. 3 4 5 



4c 



4^ 



J- -TL — 4_ &c. = 



^ bd^ ^ bdi-bd'- 



+ &c. = &c. 



which fums, with the exception of the firft, conftitute a 

 feries agreeing in form with that folved above, and from 



c d^ 



which we derive 7—-; ■ for the fum required. 



b id — ly 



Cor. — If we make a in the firft feries = o, the fum of that 

 feries will be to the fum of the latter, m d — 1 : d ; that is, 



iLf. d ~ I : d' :: r,—. \, :^7-^ tt- -'^"^ when the 



b{d— i)- b [d — ly 



(without regarding what maybe the value of S), we take 



I 



2 ■ 3 

 we {hall have 



1 I I I » n 



- + - + + - + &c. = S 

 456 



+ 



+ 



I 



I 



1.2 2.3 3.4 

 In the fame way we find 

 I I I 



4-5 



+ &c. 



I. 



1 • 3 



■+- 



+ 2 + &c. = ^. 



3.54.6 4 



On the fame principle, John Bernoulli demonttrated, that 

 the fum of the reciprocals of the natural numbers is in- 

 numcrators proceed according to tlie figurate numbers of finite. Let 

 the firlt order, viz. I, 4, 20, 35, then the fum of this 1 i i ] 1 



feries will be to that of the latter, as d : d — 1 : that is, h - 4 1 H ■; + &c. 



. , ^^' ^'^^ _ . ,' , 23456 



as</- ^ •''■'• ^J-ZYj'' Tlin^iy ~ be changed into the equivalent form 



f-'"i- + ;i+f^ + T5>i^+^" 



I 2 2 4. C 



26 la 20 30 



and 



