THE SUMS OF PERIODIC SERIES. 555 



becomes equal to \|/(w, ,v) when j^ is a positive integer; and consider the surface whose equation 

 is ^ = xl/(y, .?'). Since \M« , x) = for integral values of y, the surface approaches indefinitely to 

 the plane .vy when y becomes infinite ; or rather, among the infinite number of admissible forms of 

 \//(!/, x) we may evidently choose an infinite number for which that is the case. Now the assertion 

 made conies to this ; that if we cut the surface bv a plane parallel to the plane xz, and at a 

 distance n from it, the tangent at the point of the section corresponding to any given value of x 

 will ultimately lie in the plane xy when n becomes infinite, except in the case of singular, isolated 

 values of;!,', whose number is finite between x = and x = a. For such values the sum/(.i') of the 

 infinite series may become infinite, while ff(x) dx remains finite. The assumption just made 

 appears evident unless A„ be a function of n whose complexity increases indefinitely with its 

 rank, i. e. with the value of n. 



Since the integral of f{x) is continuous, f(x) may be expanded by the formula in a series 



of sines. Let E„ be the coeflicient of sin in its direct expansion ; so that, 



a 



.(45), 



f(x) = S^„ sin ^ , 



a 



nirx 



f{x) = 2£„ sin , 



a 



where both series are convergent, except it be for isolated values of w. Consequently, we have, 

 in a series which is convergent, at least for general values of x, 



TlTTX 



= 2(J„ - E,) sin -~~- (46). 



a 



The series (45) may become divergent for isolated values of .r, and are in fact divergent for 

 values of .r which render f{x) infinite. But the first side of (46) being constantly zero, and the 

 series at the second side being convergent for general values of x, it does not seem that it can 

 become divergent for isolated values. Hence according to the preceding article the second side 

 of the equation is the direct developement of the first side, i. e. of zero ; and therefore E„ = A„, 

 or the given scries is the direct developement of its sum, which is what it was required to prove. 

 The same reasoning applies to the series of cosines. 



It may be observed that the well known series, 



^ + cos X + cos 2,r + cos 3x (47), 



forms no exception to the preceding observation. This series is in fact divergent for general 

 values of x, that is to say not convergent, and in that respect it totally differs from the series in 

 (46). When it is asserted that the sum of the series (47) is zero except for a; = or any multi})le 

 of 27r, when it is infinite, all that is meant is that the limit to which the sum of the convergent 

 series 1 + "^g" cos nx approaches when g becomes 1 is zero, except for a? = or any multiple of 27r, 

 in which case it is infinity. 



27- It follows from the preceding article that even without knowing <l priori the nature of the 

 function f{v) we may emi)loy tlie formuhr such as (.M), provided that if «"" be the highest power 



of - rwjuired by the formula, and W^C^ the remainder in the expansion of J„, tlie series 2- (',. 



be essentially convergent. For let G„ he the sum of the terms as far as that containing ?r " in the 

 expansion of A„, those terms having the form assigned by (.'i.';), that is to say cosines like cosiiy 



coming along with odd powers of - , and sines along with even powers. Then ./„ ■= G„ + if^C,. 



