THE SUMS OP PERIODIC SERIES. 537 



3. Hence, whenever /(,i) can be expanded in the convergent series which forms the right-hand 

 side of (1), the value of A„ can be very readily found, and the expansion performed. But this 

 leaves us quite in the dark as to the degree of generality that a function which can be so expanded 

 admits of. In considering this question it will be convenient, instead of endeavourinf to develope 

 /(.r), to seek the value of the infinite series 



-2 / /(a- ) sm — -dx.sm , (s) 



a •!„ a a ^ ' 



provided the series be convergent ; for it is only in that case that we can, without further definition, 

 speak of the sum of the series at all. Now if we had only a finite number n of terms in the series 

 (3) we might of course replace the series by 



- / f(x ) {sin sin ^ sin sin ... + sm sin > da 



aJ(, \ a a a a a a ] 



(4). 



As it is however this transformation cannot be made, because, the series within brackets in the 

 expression which would replace (4) not being convergent, the expression would be a mere symbol 

 without any meaning. If however the series (3) is essentially convergent, its sum is equal to the 

 limit of the sum of the following essentially convergent series 



- Sg-" / /Or ) sm da;'.sm , (5), 



when g from having been less than 1 becomes in the limit 1. It will be observed that if (3) were 

 only accidentally convergent, we could not with certainty affirm the sum of (3) to be the limit of 

 the sum of (5). For it is conceivable, or at least not at present proved to be impossible, that 

 the mode of the mutual destruction of the terms of (3) in the infinitely remote part of the series 

 should be altered by the introduction of the factor g', however little g might differ from 1. Let us 

 now, instead of seeking the sum of (3) in those cases in which the series is convergent, seek the limit 

 to which the sum of (5) approaches as g approaches to 1 as its limit. 



4. The transformation already referred to, which could not be effected on the series (3), may 

 be effected on (5), that is to say, instead of first integrating the several terms and then summino-, 

 we may first sum and then integrate. We have thus, for the value of the series, 



- / /(.r' 2g-"sm sm \dx' (6). 



a J„ [ a a j ' 



The convergent series within brackets can easily be summed. The expression (6) thus becomes 



h |>(^'> j 'Jc^--.) , '."(f-^.) J ''^' (')• 



]\-2gcos~^ ^ +^ j-ogcos-^ ^ +g'[ 



\ a a ) 



Now since the quantity under the integral sign vanishes when g- = 1, provided cos — be 



not = 1, the limit of (7) when g= \ will not be altered if we replace the limits and a of,/ by 

 any other limits or groups of limits as close as we please, provided they contain the values of x 

 which render ,»' ± .r equal to zero or any multiple of 2 a. Let us first sujjposc that we are con- 

 sidering a value of x lying between and «, and in the neighbourhood of wliich /(i) alters 

 continuously. Then, since.r'+a? never becomes equal to zero or any multiple of 'i a within the 

 limits of integration, we may omit the second term within brackets in (7) ; and since 3> - :v never 

 becomes equal to any multiple of 2 a, and vanishes only when .i' = .r, we may take for the limits 



