194 Mr. DE morgan, ON DIVERGENT SERIES. 



For any particular series A„ - Aj + ... it is enough that A„ - JiX + ... should be a continuous 

 function of a? whose differential coefficients preserve their initial signs from a; = to x = 1. But 

 thouo-h some of them should change sign, the theorem obviously remains unaffected as to summation 

 stopping at parts of the series in which no change takes place. It is then no wonder that the 

 theorem should be so frequently true. 



Whatever value a function may have when x = 0, it is obvious that if the commencing series of 



signs, namely, those of ^0, <p' 0, &c. be h f- — i- - &c. ad infinitum, the function itself, and all 



its differential coefficients, are at the first instant in a state of numerical diminution. The reason 

 is that those whicli begin negative are algebraically increasing, while those which begin positive are 

 algebraically diminishing : this follows from the well-i<nown (but much too scantily used) theorem 

 that a function is in a state of algebraical increase or decrease according as its differential coefficient 

 is for the moment positive or negative. Adopting for convenience the mechanical idea of the differ- 

 ential coefficient representing the velocity of the function, and supposing w to be the time elapsed, 

 say in seconds, let (p x = A^ — A^ai + A„a^ - ... be a function of er, A„, A^, &c. all being positive. 

 And first let A^, A,, A.^, &c. present an unbroken series of diminutions, or A^-A^, A^ - A^, &ec. 

 an unbroken series of positive signs. Then (px begins = A„, with retardation at the rate of 

 - A, per second. But Ji, is less than A„ ; therefore this rate of retardation cannot change the sign 

 of <p X in one second, unless it receive an increase. But this there is no symptom of at the com- 

 mencement, since <b"0 is positive, and the retardation begins by being checked. Hence, if a func- 

 tion start with a differential coefficient of a sign different from its own, and numerically less, it cannot 

 change sign within the next unit of increase of the variable, without the second differential coefficient 



first changing sign. Nor can it even change sign before x becomes — ^ without a change of sign in 



(b"x previously occurring. For if the velocity had continued uniform, it would then have been 



A„ -A^ or 0, and would not have changed sign till after x =— - at least ; but since the velocity 



Aj Aj 



J 

 of retardation begins by being diminished (0"o being positive), it must make this up before ^ = -r 



if a change of sign be to take place; that is, increase of retardation must come on, or (p"x must 

 become negative. All tliis will be very plainly pictured in the curve y = <px. 



Again, if ^,a,' = ^i - A^x + ... and if A, > A.^ similar reasoning shows that <^, x cannot change 

 sign before x = 1, unless <p^"x first change sign. If neither (p"x nor (p^" x change sign from x = 

 to X = 1, then it is easily collected that A„- A^ + ... lies between A„ and A„ - A^. And if we 

 suppose y/„, A^, &c. to diminish until we come to A„, then if (j)„ x = A„ -A„^i x + ... we see that if 

 neither (pl^iX nor <p„_,x vanish before x=l, we are sure that A„_2 and A„_2- A„_ ^ contain 

 A„_2 - A„^^ + ... between them; from which it may readily be proved that the theorem is true up 

 to the last but one of the converging terms, under the preceding pair of conditions. 



The useful part of this theorem in calculation, is undoubtedly its usual truth for all the 

 apparently converging terms of the series. And we see from the above that if these converging 

 terms last up to A„, then m not being >}i, the theorem is true up to J„_,, inclusive, if neither 

 ^'I_»*' nor 'pin-f''' vanish before .r = 1 . But the theorem is not universally true even for 

 converging terms. Let (px = 3 - 2x + x^ - SOx'' + Wx" - 20x^ + ... which has three terras con- 

 verging, and is of finite divergence ; so that Poisson would admit - 8 = 3 - 2 + 1 - 20 -t- 20 - ... 

 as the limiting form of the above when .r = 1 . But - 8 does not lie between 3 and 3-2. This 

 series is the development of (3 + x - x' - 19 a'') (1 + >r)'' and its second differential coefficient will 

 be found to change sign before x = I. 



We will now look at the theorem in another point of view. Every alternating series may be 

 reduced to a case of <px - (p {x -(- 1) -H (.t + 2) - ... in which (pv is a positive function from 

 II = .r to u = CO • If this be the proper developement of ^x, then \^x + \\f{x + \) = (px ; con- 



