THE SUMS OF PERIODIC SERIES. 551 



cos ss COS 3 z IT- tts: 



— —- + h ... = , from z = to X = TT, (*!)> 



1= 3- 8 4 



which are obtained by integrating several times the equation 



sin sf + ^ sin 2 sr + ^ sin 3 sr + ... = 1 (tt - «), from z - to z = 2 tt, 



or the equation deduced from it by writing tt — z for z, and taking the semi-sum of the results. 

 It will be observed that in the several series to be summed we shall always have sines coming 

 with odd powers of n and cosines with even. Of course, by clearing the series in (24) or (30) in 

 the way just mentioned we shall increase the convergency of the infinite series in which a part 

 of /(.r) still remains developed. 



When A„ or B„ decreases faster than any inverse power of n as w increases, (as is the case for 

 instance when it is the w"* term of a geometric series with a ratio less than 1,) all the terms of 

 its expansion in a series according to inverse powers of n vanish. In this ease, then, /(«) and 

 its derivatives of all orders are continuous. 



20. In establishing the several theorems contained in this Section, it has been supposed that 

 none of the derivatives of /(.r) which enter into the investigation are infinite. It should be 

 observed, however, that if /''(.r) is the last derivative employed, which only appears under the sign 

 of integration, it is allowable to suppose that /''(•r) becomes infinite any finite number of times 

 within the limits of integration. To show this, we have only got to prove that 



/ f^i'"") *'" vvdx or / f'^{,v) cos vxdx 



approaches zero as its limit as v increases beyond all limit. Let us consider the former of these 

 integrals, and suppose that /''(a) becomes infinite only once, namely, when .v = a, within the limits 

 of integration. Let the interval from to a be divided into these four intervals to a - ^, a - ^ 

 to a, a to a + X! > " + X,' to o, where ^ and ^' are supposed to be taken sufficiently small to 

 exclude all values of x lying between the limits a - 'C and a + T' for which /''"'(r) alters discon- 

 tinuously, or for which /''(.r) changes sign, unless it be the value a. For the first and fourth 

 intervals /''(a;) is not infinite, and therefore, as it is known, the corresponding parts of the integral 

 vanish for r = co . Since sin vie cannot lie beyond the limits + 1 and - 1, and is only equal to 

 either limit for particular values of ,v, it is evident that the second and third portions of the 

 integral are together numerically inferior to /, where 



/= {y>-'(a-e)-.r-'(a-^)f + \r'^{a + ^) -/"-(„+.) J, 



the symbol A ~ B denoting the arithmetical difference of ^i and B, and e being an infinitely small 

 quantity, so that /(a - e), /(a + e) denote the limits to which /(.r) tends as x tends to the limit a 

 by increasing and decreasing respectively. Hence the limit of the integral first considered, for 

 1/ = » , must be less than /. But / may be made as small as we please by diminishing ^ and 

 Y', and therefore the limit required is zero. 



The same proof applies to the integral containing cos i/.r, and tliere is no difliculty in extend- 

 ing it to the case in which /''(a) is infinite more than once witliiii the limits of integration, or at 

 one of the limits. 



2\. It has hitherto been supposed that the function expanded in the scries (3) or (22) does 

 not become infinite ; but the expansions will still be correct even if f(x) becomes infinite any 

 finite number of times, provided that ff(x) civ be essentially convergent. Svippose that f(x) be- 

 comes infinite only when x = a. Then it is evident that we may fiml a fuiiilion of .r, /'(.i), which 

 ■.hall be e<|ual to/(.x) except when x lies between the limits « - ^ and a + ^', which shall reniuiu 



402 



/ 



