552 Mr DE MORGAN, ON VARIOUS POINTS 



that Maclaurin's theorem can never yield an always divergent development with finite coeffi- 

 cients. And, so far as functions of finite form are concerned, this is true. 



So far as is known, those functions of x which yield developments of permanent divergency 

 can be nothing but divergents of another form, for instance, definite integrals the subjects of 

 which become infinite between the limits. As an example, let 



/•» e-'dt 



(fix = / = 1 + x + 2.zr + 2 . 3. x*+ 2 . 3 . 4» 4 +... 



1 J 1 - wt 



Here, x having any positive value, however small, (fix is an integral the subject of which 

 becomes infinite when t = x'\ Those who would contend that this definite integral alwavs 

 represents infinity when x has a positive value will see a confirmation of their theorem : those 

 who suspend their opinion on this point must wait for other representation of (fix before they 

 see either confirmation or refutation. 



Should the preceding demonstration be held sound, it is clear that it may be, with 

 advantage, introduced into elementary instruction. As an instance of its effective use, take 

 <px = x (e* — I) -1 . We see beforehand that the development will be convergent up to 

 x = 27r, as is otherwise known. If my memory be correct, I have seen it stated that this 

 development is always divergent. 



It is equally obvious that (1 + e 1 ) -1 is convergent only up to x = it. 



Among remarkable consequences we find that the product of a number of series is con- 

 vergent only so long as they are all convergent. But if a division by a series, (fix, be 

 introduced, then (f)x = 0, (j)'x = co , (p"x = eo , &c. are introduced among the competing 

 equations. 



When (fix is a rational fraction, a verification may be procured with ease, if the least 

 modulus belong to a real root. Let qbx = (1 - aa?) -1 (l - jStf) -1 , whence we find 



«„ = a"+ a n -'/3+ ... +/3". 



By a well known theorem, the series is convergent up to x = the limit of a„_, : o„, if this 

 fraction finally tend to a limit. Now when a and /3 are real, a n _ x : a n does tend to a finite 

 limit, but which cannot be determinately expressed; it is a~ l or /3 _1 , whichever is nume- 

 rically the less of the two, or has the lesser modulus. This agrees with the theorem. But 

 when a -1 and /3" 1 are of the form 



r (cos 9 ± sin . -y/- l), then a„_, : a„ or (a" - /3") : (a" +1 - /3" +1 ) becomes 



sin nd : r sin (w + l) B, or r~ l (cos 9 + tan nd . sin 9)~ l . 



The ordinary theorem here fails/because there is no definite limit ; the theorem in this paper 

 settles the point. If, as I believe, tan 03 = ± ^/- 1 be a true result, the limit of «„_] : a n 

 is the same in form when a and /3 are imaginary as when they are real, namely, a~ l or /3 _> , 

 indeterminately. 



M. Cauchy's form of the theorem, though imperfect in the statement of the conditions, 

 does not affect the validity of his application to Lagrange's theorem. If y = % + axfiy, we 

 have 



' 1 — X(fi y 



