192 Mr. DE morgan, ON DIVERGENT SERIES. 



the first, and only contrary in an inverted order. And it must be remembered that, A being the 

 mean value of X, (pA is not therefore that oi (pX : thus sin'^i- has the mean value 1, not 0^. Also 

 that, when a quantity is, at one or more epochs, infinite, its mean value is not necessarily positive 

 because all its values are positive. Thus tan^a; has -1 for its mean value. The mean value of 

 sec X or cosec .r is 0. This remarkable coincidence of two modes so remote from each other of 

 determining the analytical meaning of tan co and cot os , depends at last upon £± = V-i -q^ ^^ equa- 

 tion which more writers have virtually used than have openly dared to state it. The apparent dis- 

 turbance of the law of continuity when a; = oo , as in cos' eo + sin' eo = 0, &c. is perhaps what has 

 prevented the formal recognition of these relations : nevertheless they will, it may confidently be 

 asserted, not only obtain universal reception, but finally a rational and consistent explanation. 



The following is a glimpse, perhaps, of the explanation, as applied to series. In every conver- 

 gent series, the limit of the sum of all its terms is the mean value obtained from all the summations: 

 the mean of n partial summations J,, (J, + A^, {A^ + A2+ ... + .4,,) 



\&A. + A2+ 'A,+ ... + -A„, 



n 71 n 



which, as n is increased without limit, has A, + A^ + ... ad inf. for its limit. Hence, by Poisson's 

 principle, by which I mean the assumption of the right to apply the maxim, " that which is quanti- 

 tatively true up to the limit, is true in the same sense at the limit, when the limit presents an 

 incalculable form^'' — we may assert most positively, that 1-1 + 1 - ... must be ^ whenever it is the 

 limiting form of convergency : not on the metaphysical doctrine (probably suggested by the known 

 result) of Leibnitz, namely, that we can see no reason to prefer to 1, or 1 to 0, and must therefore 



• 1 . -1 1 n 1 n + 1 

 take a mean ; but because n partial summations give the mean - x - or - x according as n 



^ " W 2 W .2 ° 



is even or odd, and the limit of both is A. At the same time it is easily proved that whenever 

 the partial summation gives recurrences in which occurs at stated equal intervals, the limit of the 

 means must be the mean of one period. 



As in other cases, the diverging series whose terms are all of one sign is not elucidated by this 

 process, which nevertheless, provided we adhere to our principle, brings out the true algebraical 

 result for series which have terms alternately positive and negative. The mean of 1, \ — a, 

 I - a + a'', &c. {n summations), is 



+ + (- 1)°+' : 



1 + a w(l + a)' «(l + a)- 



if, when n is infinite, we take (- a)" + ' as 0, the mean of the values between which we cannot then 

 choose, we have (1 + a)"' as the limit. 



SECTION IV. 



Series of alternately positive and negative signs stand upon a much safer basis than those 

 in tvhich all the terms have the same signs, aud that whether their divergence be finite 

 or infinite. 



At the very outset, namely, in the mode of finding whether the series is convergent or divergent, 

 there is every possible difference between the two species above-named, which we may term 

 progressing and alternating. The progressing series d) (l) + ^ (2) + ... is convergent when the 

 first of the set 



^"""•^f:^' ^' = I°g^(^o-l)> P.= log log a^(P, -])... P„=(log)".r(P„.,-l)... 



