SERIES. 



2 



+ 



I 



7 



In a fimilar manner we have 

 I I 



+ --- + -^ = S" = 



I 



— + 



I- 



5' '^ r ~ ^' 



7T + 3. ,j, 



+ 



and on the fame principle, the fum of the feries, 



III. 



I + ■ — + ■ — + - — + occ. 



^ 3=" 5'" 7" 

 may be found, n being any pofitive number whatever. 



Euler's method is ilill more general than Landfii's, but 

 It depends upon principles of very nearly the fame origin : 

 we muft, however, limit ourfelves to giving a lew of the 

 principal refults. Reprefenting by ^ the femi-circumference 

 to radius i, he (hews that 



I 1 



+ ? + ? 



+ 



&c. — 



I 4- 



where the law of the firft multiplier is obvious, but the co- 

 tfficicnts of -', X , &c. are not fo eafily feen : thofe for the 



z c 6qi 



following powers arc - -% - - rr", t", &c. 



^ 5 3 loS 



If each of thefe feries be multiplied by their firft fraftion, 

 they give 



I 



I I 1 „ 



— + — + >- + &c. = 

 2' 4" 5 



I 



&c. 



4* "*" 6* 



&c. 



+ &c. = - 



2'" 

 1.2.3 



2' 

 7777.5 

 &c.; 



3 



and fubtraAing thefe from the firft, we have 



I 



+ &c. 



+• 



3' 

 &c. 



+ 



5' 



Sec. 



&.C. 



2' — I 



2' 

 2'- 1 



2' 

 &C. 



2-3 



• ■ > 



I 

 I 



I 



3 



Again, fubtrafting the firft from thefe lalt, we find the 

 fum of the powers under the alternate figns plus and minus, 

 and fo on, almoll in cndlefs variety. 



Other feries, wliofe fums are found in nearly the fame 

 manner, are as follows, viz. 



For a great variety of other ferles of this kind, fee 

 Euler's " Introduftio in Analyfin Infinitorum," and his 

 " Inftitutiones Calculi Differcntalis." See alfo Spence's 

 " Effay on the Tlieory of the various Orders of Logarithmic 

 Tranfcendents," 410. 1809; in which feveral feries, fome- 

 what fimilar to the above, but which were not fummable by 

 Euler's method, are treated of, and inveftigated in a very 

 able manner. 



We ought perhaps to apologize to fuch of our readers 

 who are not interefted in mathematical enquiries, for the 

 length to which we have extended this article ; but thofe 

 who are, will not, we prefume, be difpleafed to find in a con- 

 denfed form a general view of the firft introduftion, andluc- 

 cefiive improvements, which have been made in this import- 

 ant branch of analyfis. Wc have, of courfe, been obliged 

 to pafs over in filence many authors who have written on this 

 fubjeft ; but we have endeavoured to include all thofe who 

 have introduced into the dodrine any methods diftindlly dif- 

 ferent from thofe who preceded them, at lealt, if we except 

 Mr. Spence's method, publiftiedin his" Logarithmic Tranf- 

 cendents," and that of M. Arbogaft, given in his " Calcul des 

 Derivations." We had indeed, in the firft inilance, intended 

 to give an illuftration of the principles of thefe two authors ; 

 but the length to which the article has already extended, and 

 the nature of their notation, which render necell'ary a confider- 

 able degree of previous explanation, put it out of our power 

 to execute this part of our plan, and we can therefore do 

 nothing more than refer the reader for information to the 

 works themfelves ; we refer him alfo to the " Calcul des 

 Diflerences Fines," by La Croix, and to an ingenious me- 

 moir by profeftor Vince, in the 7 2d volume of the Pholofo- 

 phical Tranfaftions. 



As the preceding article is arranged wholly with refer- 

 ence to the hiftorical order of the fubjeft, we intend, in coa- 

 clufion, to furnifh the reader with a general fynopfis of the 

 doftrme of frries for the advantages of pradical operations. 

 12. General Syntpfu for the Summation of Series. — In the 

 following tablet, S denotes the fum of a finite number of 

 terms (n), and S the fum of an infinite number. 



1. To find whether the fum of any propofed feries be 

 finite or infinite ; let p, q, r, be any three equidiftant terms ; 

 then, if/> [q - r) > r (p — q), the fum is finite ; but if 

 P {q — r) <~. r (p — q), it is infinite. 



2. The general term of a feries, when any order m of its 

 differences vanifti, is of the form, 



T = A n "• + B n""-' -f C n™- ' + &c. 



and its fum of the form, 



X = A'n"' + ' + B'/i'" + dr-' + &c. 



the values A, B, C, &c being found as ftated in art 5, and 

 thofe of A', B', C, Sec. in a limilar manner. 



3 Simple arithmetical Series. 

 a +(«+</) + (<'+ 2r/) + (a + 3^) .... fl -t- (n - 1^ J 



(v) Infinite (S) = - (2fl-|- (h - l)(!]. 

 If the feries decreafc, then </ is negative, and 



(S) ="^ (2 <,-(«- l)d). 



4. Simple geometrical Seriet. 



a -(-ra-f-r'a + r'a. . . . r' ' a 



(.-) = -"- . (8) = .^::-^- 



^ ' I — r ^ ' r — I 



5. P»<wert 



