192 Mr DE morgan, ON INTRODUCING DISCONTINUOUS CONSTANTS 



7n + 



l-«-' 1 — «-' 1 — «" rt — 1 ^- + 1 



and resolved into series of positive powers of a and .r, with a similar 

 subsequent process, it yields 



\ - a \ - a \ - a '" (1 - a) (1 + a;) ' 



Let us now apply this method to examine some of the more 

 common series of analysis : let us take 



1 + rt + a- + rt^ + ••■ 



Multiply every term by x, and it will then appear to be 



U.X vx, where ax = ax, 



va'^X ' 

 and nx and vx are any particular solutions of 



0ar = a; + ^ {ax), <px =.(p {ax), 



let fix = , vx = C, 



X a^x 



then the series is 



1 - rt 1 — rt 



Like all other results of strict methods of passing from the sum 

 of w terms to the sum of an infinite series, this expression is infinite 

 when tlie series is infinite. But my object here is to remark, that 

 owing to a'"x = x having only the root x = 0, there can be no dis- 

 continuity among the values which correspond to arithmetical values 

 of the above series. 



If we consider the series 



rt° rt"* 



as a case of 



+ .^. ,w^ . o^ +^•• 



^ X + 1 ' {x+l){x + 2) 

 we find the value of the preceding to be as follows, 



