THE SUMS OF PERIODIC SERIES. 563 



For in this case the series (67) is more rapidly convergent than {66), and therefore its 

 convergency remains finite. 



Cor. 2. If the series (66) is essentially convergent, and if the terms of (67) are derived from 

 those of (66) by multiplying them by the ascending powers of a quantity g which is less than I, 

 and which becomes 1 in the limit, the limit of V is equal to U. 



It may be observed that when the convergency of (67) does not become infinitely slow when h 

 vanishes there is no occasion to prove the convergency of (66), since it follows from that of (67). 

 In fact, let V„ be the sum of the first n terms of (67), U'„ the same for (66), V„ the value of V for 

 h = 0. Then by hypothesis we may find a finite value of n such that V — V„ shall be numerically 

 less than e, however small h may be; so that 



V = V„ -I- a quantity always numerically less than e. 



Now let h vanish : then V becomes F„ and V„ becomes U„. Also e may be made as small as we 

 please by taking n sufliciently great. Hence U'„ approaches a finite limit when m becomes infinite, 

 and that limit is Fq. 



Conversely, if (66) is convergent, and if f7 = Fq, the convergency of the series (67) cannot 

 become infinitely slow when h vanishes. 



For if n'„', F„' represent the sums of the terms after the w"' in the series (66), (67) respectively, 

 we have 



whence F„' = V- U- (V„ - U„) + U,'. 



Now V ~ U, V„ — U„ vanish with /;, and f7„' vanishes when n becomes infinite. Hence for a 

 sufficiently small value of h and all inferior values, together with a value of ra sufficiently large, and 

 independent of h, the value of F„' may be made numerically less than any given quantity e however 

 small ; and therefore, by definition, the convergency of the series (67) does not become infinitely 

 slow when /* vanishes. 



On the whole, then, when the convergency of the series (67) does not become infinitely slow 

 when h vanishes, the series (66) is necessarily convergent, and has F„ for its sum : but in the 

 contrary case there must necessarily be a discontinuity of some kind. Either V must become infinite 

 when h vanishes, or the series (66) must be divergent, or, if (66) is convergent as well as (67), 

 U must be different from F,,. 



When a finite function of x, f(x), which passes suddenly from M to iV as a? increases through a, 

 where a > a > 0, is expanded in the series (15) or (30), we have seen that the series is always 

 convergent, and its sum for all values of x e.xcept critical values is f(x), and for x = a its sum is 

 A (M + N). Hence the convergency of the series necessarily becomes infinitely slow when a - x 

 vanishes. In applying the preceding reasoning to this case it will be observed that /t is a — x, 

 F„ is M, and f/ is ^ (M + N), if we are considering values of a? a little less than a ; but h is x — a 

 and V„ is N, if we are considering values of a/ a little greater than a. 



When the series (66) is convergent, as well as (67), it may be easily proved that in all cases 



U=V,- L, 

 where L is the limit of F„' when h is first made to vanish and then n to become infinite. 



40. Reasoning exactly similar to that contained in the preceding article may be applied to 



F(x, h) d,v is a convergent integral, 



a 



we may say that the convergency becomes infinitely slow when h vuni.she.s, wlieii, if ..V be the 

 superior limit to wliich we must integrate in order that the neglected part of the integral, or 



