Mr. DE morgan, ON DIVERGENT SERIES. 193 



which is not equal to unity is greater than unity ; and divergent when the first which is not 

 equal to unity is less than unity. But <p {^) - (p (2) + ... is necessarily convergent, provided 

 only that <p(co ) = continuously, or that the terms ultimately diminish without limit. 



A progressing series must be either convergent or infinitely divergent ; an alternating series may 

 be convergent, or either finitely or infinitely divergent: but the infinite divergence of the latter 

 is of a different character from that of the former. I very much doubt whether it is quite 

 correct to apply the same phrases to both kinds of series. 



It is easy to apply Poisson's principle to alternating series, even when they are of infinitely 

 diverging form. We can always contrive to find positive quantities B^, 5,, &c. in such a 

 manner that A„B^- A^B^ + A„Bi- ... is convergent, up to a certain value of a variable contained 

 in fi„ &c., which makes them become severally = 1. Thus !-« + «=_ is a limiting form of 

 1 - ax'" + a^ic'"' — a^a;~"' + ... which, n being > 1 is certainly convergent down to a; = 1, exclusive; 

 and this whatever the value of a may be. Whether this limiting form is always (1 + a)-' may be 

 a question; but, as I think is sufficiently shown in various parts of this paper, the question may 

 also be asked about the finitely diverging series which have been so confidently allowed. 



When an alternating series is convergent, and a certain number of its terms are taken as 

 an approximation, the first term neglected is a superior limit of the error of approximation. This 

 very useful property was observed to belong to large classes of alternating series, when finitely or 

 even infinitely divergent : I do not remember that any one has denied that it is universally true, 

 while many have implicitly asserted it. When the series is convergent for a certain number 

 of terms, particularly if the terms become very small before they begin to increase again, it obviously 

 makes the divergent alternating series practically as useful as the converging series, perhaps even 

 more so, for it is very frequent that the greater the ultimate divergence, the greater also is the 

 primitive tendency towards convergence. 



In any series Po- P^ + P.^ - ... this theorem is obviously true as long as the remnant 

 P:, ~ Pn + \ + ••• has the same sign as P„, or the positive sign. Thus, if /"„ - P„ , + ... = Q„, we 

 have for the series P„ - P, + Q., and Pg- P^ + P., - Q^: if Q, and Q^ be positive, the series is 

 greater than P,, - P, and less than P„- P^ + P.^; which is a case of the theorem. It is also clear 

 that if either Q^ or Q^ be negative this case is not true. 



That the theorem is not universally true will appear in the following instances : 



i = cos^ a — cos^ 2 a + cos'' 3 a - ... 



3t + #- -3i' + t* - 3t'' + ... 



It is not true that 1 always lies between cos' o and co.s^ o - cos'' 2 a, or that (1 - 3t) (1 - f)-^ 

 always lies between 1 and I - 3t, whenever t is positive. The following investigations, thouo-h they 

 will fully explain why it is that the theorem is so often true, are insufficient to distinguish accurately 

 between those in which it is and is not true. 



When (px can be expanded into A - Bx + C.v' - ... (A, B, &c. being positive), we take 

 the known form 



00 + (p'0.a.+ <p"0- + + 0'"'O — + 0'"+"(e.i;) 



2 .3...n ^ 2.3 



in which 6<l. If then <p'o, (f>"'0, 8ec. be negative, and cpo, (p"(), he. positive, and if (pw, (p' x, &c. 

 each preserve, up to ,r = a, the sign it starts with when x = 0, there is no question that tlic theorem is 

 true from .r = to .r = a. Thus common differentiation with respect to ,v will prove the theorem 

 for the case of 



/■ 



-dv 



= 1 - a; + 2j?- - 2.3.r' + 



1 + ajv 



