^yi8 Mr, Marrat on Neutral Series. [June, 



greatest limit; the other hmit is zero, as we shall soon demon- 

 strate. <■ 

 Let us reduce into series, 



J- = 1 - a; + a« - JT^ + j:-* - a' + &c to infinity. 



- — ! — i = 1 — X- + JT^ — X* + x*' — a7 ^ 0,9 — &c 



I + a + x' 



= 1 — x 4- X* — a:* + x" — jr» + x'2 — Sec 



i ~ = 1 _ j: + x* - a*' 4- a '0 - x'' + a:'* - &c 



The first of these series contains the successive powers of a, 

 or it may be said to contain all the terms ; and when x = 1, it 



becomes j— = i.= l— 1 + 1 — 1+ &c. to infinity. 



The second series wants several of its terms, and on that 

 Account it ought to be less than the first; howeve<*, when a: = ], 

 .it becomes -i-=l-- I+l— 1 + &c to infinity. 



In the third series, more of the terms are wanting, and this 

 .series should be less than the second series; however, when 

 2" = 1, it becomes ^=1 — 1 + 1--1+, &c. .... to infinity. 



In the fourth series, several more of the terms are gone ; but 

 when X = 1, we have 4- = l — 1 + 1 — 1+ &c to infinity. 



It appears from all these expressions that the two first terms 



remain constant, but that the other terms gradually disappear, 



according as the number of terms in the denominator increase ; 



whence it also appears from the law of continuity, that the value 



.of the series ought regularly to diminish. 



We will now determine the series, when the denominator con- 

 . gists of an infinite number of terras ; thus let 



-■: — -4 T-^-rr = A + Bx+Cj:^ + Dx3 + 8cc 



Multiplying the second side of this equation by the denomi- 

 nator of the first, we have 



A + B X + C a:* + D x' + &c. 



+ Ax + Bx^ + Cx' + 



Ax^ + Bx' -^ .... 

 + Ax" + .... 



and equating the coefficients of the same powers of x, A = 1, 

 B + A=0, C+ B + A=0, D + C + B + A=0; that is, 

 A = 1, B = - I C = 0, D = 0, &c. ; therefore, 



: — - — , . ^ ■ = 1 — X, where all the terms, except 



1 •♦-*•+ jr^ + x* + &c. ad infin. ' ^ 



the two first, or 1 — x, have vanished. Let x = 1, then 

 ! = 1 — 1 = 0; whence it is manifest 



i — I + I — I + &c. to infinity 



that when the denominator consists of an infinite number of 



