Mb. DE morgan, ON DIVERGENT SERIES. 201 



incidental to one particular process ; we were not bound to Legendre's integral ; and this dis- 

 continuity disappears in the result. But the second is essential to the problem ; the series 

 satisfies a certain differential equation, the complete solution of that equation is ascertained, and 

 therefore the series mttst be represented by its equivalent solution of the equation. No other 

 equivalent could have been anything but the one we found, or the same in a different form. 

 As matters stand, then, we cannot have a continuous relation between the series and its invelop- 

 ment : and this, I will venture to prognosticate, will continue until the definition of integration is 

 extended. 



Let us now try ... + -z- rr + ... which call S' -j . 



Proceeding just as before, and, 6 lying between mc and {m, + l)c, we shall find, m being m + X, 

 as before, 



-b)w — (m'c— 6)0 



'^'l + (6 + ^e)^ = (-'^"'/ 



sin ydu. 



1 + (h+pcY ^ ' J^ e'" +e- 

 But Legendre has also shown the following, g being not greater than h: 



f 



(e^S-e'^Msin — 

 sin vdv = 



"" h Z ,1 ■Kg 



e* + e * + 2 cos — ^ 



whence the series in question is 



(e'- e ')&m [m + ^ --\ir (e'-e ^cos — 



\ cl 27r C 



or 



£<■+£ '■+2cos2m + l Tf e^ + e <:-2cos^ 



27r(- I)" 



Zira 2it / 



e <■ + e" "■ + 2 cos I 2 m + 1 - 



c ! c 



which is, as it ought to be, a solution of \^(6 + c)= ~ \^b. Now consider the series, 



^ " 1 +6* ~ r + (6 + cf "^ 1 + (6 + Zcf ~ '" 

 which IS a solution of y + - 



dx° \ +e" 

 cos* e"^ da r' smwe"da! 



, ■ r cos* e ax r 



whence y = sin x / cos w I 



1 + e 



on which may be repeated all the remarks on the preceding case. But in this case, when c = 0, 

 the value of y gives, as it should do, ^e"{\ +6^)"'. 



The danger of integrating over a diverging series is thus shown to be incident to alternating 

 as well as progressing series. It cannot be denied that Poisson has separated the only case in 

 which integration can be used with some freedom and safety on non-arithmetical series : namely, 

 the finitely diverging series which lies between the convergent and divergent cases. Whether 

 the freedom is entire and the .safety absolute is more than can be determined at present : 

 unacquainted as we are with all the varieties of the discontinuity which appears in limiting cases 

 of integration, as now understood. On this point, I must refer to the preceding part of this 

 paper. 



With regard to the alternating double series ■{■ A _.iai''- A _iX'^ + A^- A^x + A^x^ - ... we 

 now learn tliat, whenever complete continuity exists, A - A^x + ..., x being infinite, must have 

 the same value as A ^^-x'^ - A _.^x~^+ ... when .r"' is nothing; that is, must vanish, generally 

 speaking. This observed tendency of A - A^x + ... has been already noticed. 



