THE SUMS OF PERIODIC SERIES. 541 



and evidently 



limit of 2 - g" sinnz = 0, when ^ = 0, 



that is of course supposing z first to vanish and then g to become I. Also the limit of 



2 - g'smnz changes sign with x, and recurs when z is increased or diminished by Sir. Hence, 



the series (12), which has been proved to be convergent, is in all cases the limit to which the sum of 



the convergent series 2 - ^" sin nz tends as g tends to 1 as its limit. Now the series (11) mav be 



decomposed into two series of the form just discussed, whence it follows that the series (3) is 

 always convergent, and its sum for all values of x, critical as well as general, is the limit of the 

 sum of the series (5), when g becomes equal to 1. 



The examination of the convergency of the series (3) in the only doubtful case, that is to say, 

 the case in which f(x) is discontinuous, or does not vanish for a; = and for x = a, is more curious 

 than important. For in the analytical applications of the series (3) it would be sufficient to regard 

 it as the limit of the series (5) ; and in the case in which (3) is only accidentally convergent, we 

 should hardly think of employing it in the numerical computation of f{x) if we could possibly 

 help it, and it will immediately appear that in all the cases which are most important to consider 

 we can get rid of the troublesome terms without knowing the sum of the series. 



The proof of the convergency of the series (3) which has just been given, though in some 

 respects I believe new, is certainly rather circuitous, and it has the disadvantage of not applying 

 to the case in which /'(a?) is infinite*, an objection which does not apply to the proof given by 

 M. Dirichletf. It has been supposed moreover that/"(,^?) is not infinite. The latter restriction 

 however may easily be removed, as in the end of the next article. 



7. Let /(.r) be a function of x which is expanded between the limits x = and x = a m the 

 series (3). Let the series be 



, . ttX . 2 rrx rlirx 



At sin — + A., sin ... + ^„ sin + ... , (15), 



a a a 



and suppose that we have given the coefficients ^,, A2..., but do not know the sum of the series 

 /(,r). We may for all that find the values of/(o) and /(a), and likewise the values of x for 

 which /(.(■) is discontinuous, and the quantity by which f{x) is increased as .r increases through 

 each of these critical values. 



For from (9) and (10) 



«^„ = - (/(<0 - (- !)"/(«) +SiN- M) cos "^ + - , 



R being a quantity which does not iiccome infinite with n. If then we use the term limit in an 

 extended sense, so as to include quantities of the form C cos wy, (of course C(- I)" is a particular 

 case,) or the sum of any finite number of such quantities, wc shall have for « = 95 , 



limit of nA„ = - |/(0) - (- 1)'7(«) + ,S'(iV"- A/)cos^^i. ... (l(i). 



• Thin rMlhction may however be diapcnneiJ with l>y what i« proved in Art. 21). f Crclle'» ■/ourna/, Tom. iv. p. 157. 



Vol.. VIII. Paht V. .J A 



