THE SUMS OF PERIODIC SERIES. 5*7 



We get by integrating by parts 



]/(.•■") sin dx ■■ /(.r)cos + — /(.r)sin + — / (.r)cos— ... 



a utt a \mrj ■> ' a \mrJ a 



o 

 Multiplying now both sides by - , and taking the limits of the integrals, we get 



^"= '-^r ^f^'"'' - (- '>"-^(°)^ - -• (-) V"(0) - (- l)"/»} + -> - (27)- 

 a Tltr a \nTrl 



Hence, if n be always odd or always even, A„ can be expanded, at least to a certain number of 

 terms, in a series according to descending powers of n, the powers being odd, and the first of 

 them - I. The number of terms to which the expansion in this form is possible will depend on the 

 number of differential coefficients of /(«) which remain finite and continuous between the limits x = 

 and X = a. Let the expansion be performed, and let the result be 



■'if, = Of) - + O, — + Of — + ... when n is odd ; | 



n n n 



(28). 



A„ = Ef, — i- E, ■— -k- Ei —r + ... when n is even. 

 71 ' nr nr J 



Comparing (27) and (28), we shall have 



/(O) = ^ (0„+£„), /(«)= ^ (0„ -£„),! 



i 4 



rO^) =-~ (». + E,). /'(«)=- :^(0. - E,),\ (29), 



and so on. The first two of these equtions agree with (18). 



If we conceive the value of A„ given by (27) substituted in (26), we shall arrive at a very 

 simple rule for finding the direct expansion o(f''{x). It will only be necessary to expand A„ as 



far as — -j , admitting (-1)° into the expansion as if it were a constant coefficient, and then, sub- 

 tracting from A^ the sum of the terms thus found, employ the series which would be obtained by 

 differentiating the equation (2i) fi times. It will be necessary to assure ourselves that the term 



in — vanishes in the expansion of .4„, since otherwise /''(a?) might be infinite, or/''"'(.r) discon- 

 tinuous without our being aware of it. It will be seen however presently (Art. 20) that the 

 former circumstance would not vitiate the result, nor introduce a term involving n"''. 



Should A„ already ai)pear under such a form as - + c"; (-!)"—;-(• ra"c", &c., where c' <\, it 



n w 



will be sufficient to differentiate equation (24) /u times, and leave out the part of the series which 



becomes divergent. For it will be observed that the terms e", w'r", in the examples chosen, 



decrease with - faster than any inverse power of n. 

 n 



14. Let us now consider the odd differential coefficients of f{x), supposing f{x) to be 

 expanded in a series of cosines, so that 



