190 mh. de morgan, on divergent series. 



2 f (,r"+^+ — I cosMat) - (a" + ' H —]cos(n+l)av 



V^tt/, 5 e '^dv 



" " 1 - 2a? cos at! + as' 



1 /•+»(>r" + ' + .»"<""^'') cosMO«-(«" + .r"'') cos (w + l)au _, 

 v/ir-'-oo (« + .r )-2cosau 



= / 2_\ '_ — _a_;^ i co%nav e' dv . 



y/ir J_„ f (*• + >t;" ) - cos ffli) 



The series continued ad infinitum is expressed by the value of the integral just found, in 

 which n is made infinite, being the very remainder which is called nothing in the original and 

 fallacious pi-ocess. Many interesting forms might be derived from the preceding and similar 

 cases, but having no reference to the subject of this paper. 



When the terms of a divergent series separately vanish, the series having remained divergent 

 up to the time of evanescence, it is customary, in elementary works, to assume that the series 

 itself vanishes: or + +0 + ... is taken to represent 0. Very frequently, no doubt, the in- 

 velopment shows that this is correct ; and I think I shall be able to show that if the function 

 be perfectly continuous on both sides of the epoch of evanescent form, a reason can be given why 

 it must be so. But so far as the series itself is concerned, we have no right to come to such 

 a conclusion, unless we can shew that as the evanescent form is approached, the invelopment 

 diminishes without limit. The following instance will show the necessity of this caution. 



^« £-"''' cos ht dt , 

 The integral / ^ is convergent for all values of a, however small, and cer- 

 tainly is not comminuent with a, but approaches the limit ^7re~, the well-known value of 

 f" cos bt {i + f)'' dt. Expand the first side into 



n + 2 . 



J^ e-'''-'cosbt(l-t' + t'- + (-i)»r-H(-l)» + '^-^- J 



which, from 



/ 



".-"cosi^ f"di r(2n-H)cos{(2n + l)tan-'(fta-')} 



gives, making tan "' (6a"°) = 



(6^ + a')--5cos0- 1.2(6= + a'')-8 cos 30 -h 1.2.3.4(6' + a*)-*cos 50 -, 



Sn + l - €'"''00561 ^"'^^dt 



+ (- i)"1.2.3...2n{h' + a*) a cos (2re + 1) + (- 1)" + ' / '^ . 



If we neglect the last term, or suppose n infinite, we have expanded the given integral into a 

 divergent series of which all the terms are comminuent with a : for a = gives 9 = Att. When we 

 have the remainder, we may, by retaining its proper value, allow the preceding form 0^-0 + + ... 

 to stand for : but otherwise the appearance of that form must be a warning, when it arises from 

 the value of a divergent series, that there may be some finite equivalent which is not to be neglected. 



It is worth noting that immediately before the terms of the preceding series vanish, thev are all 

 of one sign, or cos0, cos 3 0, 8ec. are of alternate signs. This is one out of the constantly recurring 

 cases in which it happens that the difficulties of series are mostly incident to the divergent case in 

 which all the signs are the same; the illustration of which is the subject of the next section but one. 



