Mr. DE morgan, on DIVERGENT SERIES. 203 



From what precedes, we are warned to expect some discontinuity arising in the treatment of 

 any series (p{x) J= (p{j[t + 1) + ... i{ (p(a; + n) = (p(,v - n), unless that series be an analytical 



equivalent of 0. And even in this latter case, it is to be remembered that — is the real form, and 



that when -^w = 0, there may arise cases of exception in which the series represents a finite quantity, 

 and even infinity. This particular point has been so beautifully illustrated by Poisson, in his 

 treatment of the series ^ + cos + cos 2 + ... that nothing is left for any one else to say, at 

 present. 



In mentioning once more the name of tliis distinguished analyst, I may state that the point in 

 which I have freely ventured to question his judgment is not as to the wisdom of the course he 

 took, in rejecting divergency from the integral calculus as he found it, but as to the grounds on 

 which he asserted a final and fundamental difference between what he adopted and what he rejected. 



A. DE MORGAN. 



University College, London, 

 January 15, 1843. 



ADDITIONS. 



Page 192, line 8. It is not asserted that cos' co + sin^ co = 0, for the mean value of each of the 

 terms is -i, and cos^ so + sin^ 03 = 1. Many errors may be made by forgetting that (f> sin a;(x = co ) 



r-2n 



or <f> (0) is not the mean value of (f> sin x, but / (p (sin x) dx ■— 2w. 



•'0 



Page 201, last four lines. If it should seem for a moment that this reasoning would apply 

 equally to Au + A^x + ... and — A^iX'' - A .2^~^ - ..., remember that the theorem in Section IV 

 (to which the exceptions are only occasional) shows that A_iX~^ — A_2X'^ + ... lies between 

 A_^ ir"' and A_^ a;"' — A _2!i!'^ when a is great : but that we have no such argument from which to 

 infer the comniinuence of — A_i a?~' — A_2 •'"'^ — ••• and x~^. Still however, the equality of this 

 last to A^ + Ai X + ..., when there is no discontinuity, would enable us to predict the very large 

 number of cases in which A^ + Aiiv + ... is infinite and negative when x is infinite and positive. 



Vol. VIII. Vaui II. Do 



