INTO ARITHMETICAL EXPRESSIONS FOR INFINITE SERIES. 193 



fix = a', yx = —— , (px = «' + —1— (h{x + 1). 



A particular solution of (px = ^^"^ "^ ^^ is ux = T (x + 1) 



.r + 1 V i / 



No more simiDle value of fxx can be found than the finite integral of 



fiX a' 



1.2.3...X 1.2.3. 



..x 



If fix could be expressed, the value of the preceding series would be 



^"- 1.2.3...oc (l-^-^--^)- 



What we have here to observe is, that in consequence of ax = x + l, 

 arx increases without limit for all values of x, and there is no dis- 

 continuity. 



I shall only further remark, that the preceding results confirm, so 

 far as they go, an opinion which I have long entertained, namely, that 

 series which may be divergent, or which may be brought as near to 

 divergency as we please, such as that for e", require much less cir- 

 cumspection than those which can never be made to diverge. In the 

 first, generally speaking, the arithmetical value (between the limits of 

 convergency) is the analytical value throughout; in the second, there 

 is frequently discontinuity in the arithmetical values, and the general 

 equivalent of analysis is not easily expressed. I will not however 

 enlarge upon so general a topic, but beg to remain. 



Dear Sir, 



Yours very truly, 

 .,, ^ , AUGUSTUS DE MORGAN. 



,•) Upper Gower Street, 

 April 30, 1836. 



P. S. Some time ago, I communicated to the Society what I con- 

 sider a failure in the proof of the celebrated theorem of M. Abel, on 

 the expressibility of the roots of equations which are values of a 

 periodic function. As I have since printed my objection in the Cal- 

 culus of Functions alluded to in the preceding paper (^. go, 302, 303,) 

 I take this opportunity of referring to the subject. 

 Vol. VI. I>Aur I. Bb 



