SERIES. 



r>nd let this laft be refolved into the infinite feries 



1 I i I I . 



-J 1 1- 1 h &c. — I 



2 6 12 20 30 



I I 1,1^ ' 



-{ u 1 +&C. = - 



6 12 20 30 2 



I I I „ I 



+ — + — + — + &c. = - 



12 20 30 3 



II. I 



+ - - + — + &C. = - 

 20 30 4 



1 „ I 



+ — + &c. = - 

 30 5 



+ &c. = &c. 



Whence it follows, that the fum of 



1 I 



— 4 



2 3 



+ 



I 



+ — + 



2 



+ 



+ &c. ad irifinitum = 



■A 1 L -— -|- &c. ad Infinitum 



456 



which equality can only have place when the firft fum is in- 

 finite. 



John Bernoulh afterwards found the fum of the feries of 

 the reciprocals of the natural fquares, a problem mentioned 

 by his brother, in his fcholium to propofition 17, in which 

 he declared that the folution of it had evaded his induftry ; 

 and that whoever folved it fhould receive his warmefl: 

 thanks. 



It (hould be obferved, however, that though James had 

 failed in finding the true fum, he had difcovered feveral cu, 

 rious properties of this feries ; viz. that the fum of the odd 



terms, i H 7 -\ : + - ^, is to the fum of the even terms, 



1 



"2^ 



-^ ^, as 3 to 1. And generally, if we have a 



8 



feries of the reciprocals of any powers whatever, as — ^ 4- 



. 1 1 4- &c. the fum of the terms in the odd 



2" ^ 3" ^ 4- ^ 



places beginning at unity, is to the fum of the terms in the 



even places, as n' — i is to i. Hence, 



I I t „ II 



I + - i + — +■ - - + &c. : -- + — 4- 



3' r T 2' 4' 



63 



+ &C. :: 7:1. 



John Bernoulli's folution of the above problem depends 

 upon the expreflion for the fine of an arc in terms of the 

 arc, the fame as that of Landen, of which we fliall fpeak in 

 the fubfequcnt part of this article, and fhall, therefore, only 

 give here the refults that Bernoulli drew from his folution ; 

 viz. he proved that 



I I I ' I , 'r' 



I I 



? + "3 ■ ■ 4° ■ 5" ■ 945 



&c. &c. &c. 



denotes the femi-circumference {)f a circle whofe ra- 



3^-^^-*- 



— -f &c. = — 

 5' 90 



I + 



I I I - 



where x 

 dius is ( 



Montucla has, by mirtake, attributed the firll 



fummation of this feries to Euler, fee page 209, torn. iii. 

 " Hiftoire des Mathematiques." 



We fhall only further obferve with regard to thefe au- 

 thors, that we here find the firft notice of continued expref- 

 fions of the form 



^' ^ -t- •/ a -t- 



+ .V a 



&c. 



+ b V a + b ^-J 



+ b ^ a 



+ &c. 



with the method of fumming them by means of quadratic, 

 cubic, and biquadratic equations. See our articles Quadra- 

 tic, and Surds. 



3. Montmort's Method of Series.— Th^ two methods above 

 illuftrated, by means of which the Bernoullis arrived at the 

 fummation of various feries, are both indirect, and are belter 

 fuited to finding fummable feries, than to the fummation 

 of any feries propofed ; they are moreover only applicable 

 to luch feries as continually decreafe ad infinitum. 



In 17 12 another intcrelting correfpondence took place 

 on feries of a different kind, between M. Montmort, John 

 Bernoulh, and his nephew Nicholas Bernoulli. They were 

 led to thefc coufiderations, in confcquence of certain pro- 

 blems relating to the dodtrine of probabilities, which at that 

 time began to excite great interelt amongft both the Enghfh 

 and French mathematicians. The objeft here was not the 

 determination of the fum of an infinite number of dccreafing 

 terms, but tlie fummation of any finite number of terras, 

 either mcrcafing or decreafing ; and the formula of M. 

 Montmort, given at page 65 of his " EiFai d'Analyfe fur 

 les Jeux de Hazard," fecond edition, for this purpofe, is as 

 follows. 



Let a + b + c -ir d + e + f ■^- &c. be the propofed 

 feries, and n the number of terms whofe fum is required ; 

 alfo, let D;, D", D'", D", &c. be the firll terms of the firft, 

 fecond, third, fourth, &c. dift'ci-eiices ; then will the fum of 

 the H terms be exprefled by 



„a + "J'Ll-^ D' f "^"-') ("-^) D" 

 1.2 1.2.3 



4- 



("-3) 



1.2.3.4 



4- &c. 



which feries will terminate in all cafes where any of the 

 order of differences become zero ; but in others it will only 

 give an approximation. 



Let it be required, for example, to find the fum of n 

 terms of the natural feries of the fquares 



Here 

 therefore 



I' -r- 3' 4- 3' + 4' 4 5- . . . . »■■ 

 a = I, D' = 3, D" = 2, D'" = o ; 



n (n — l) n (n — l) (n - 2) 



n 4 — ^ ' 3 -I- — i ^-i '- 



1.2 1.2.3 



is the fum required. 



If it were the feries of triangular numbers, 



I I- 3 4 6 4 10 -(- . . . . n -' 



then we fhould have 



a= I, D'= 2, D"= I, D"' = o; 

 therefore the fum of n terms will be expreffed by 



1.2 1.2.3 



N n 2 



Fronr. 



