THE SUMS OF PERIODIC SERIES. 545 



when g from having been less than 1 becomes 1, is /(.r), x being supposed not to lie beyond the 

 limits and a. For values, however, of x for which f(x) alters discontinuously, the limit of the 

 sum is the arithmetic mean of the values of f(x) for values of x immediately above and below the 

 critical value. I assume this as being well known, observing that it may be demonstrated just 

 as a similar theorem has been demonstrated in Art. 4. 



10. Let us now consider the series 



1 /-<! , 2 /-I , tiTTx' mrx 



- / f(x)dx-\ — 2 / /(,r)cos da;. cos (22). 



a J^ a J^ a a 



We have by integration by parts 



2 r., ,. n-TTX , 2 mrx 2 a U'ttx' 2 a r , mrx , 

 - fix ) cos dx = fix ) sin + ——„ fix) cos f (ic) cos dx ; 



and now, taking the limits properly, and employing the letters M, N, a and S in the same sense 

 as before, we have 





WTT.r , , 2 „^,^ ,,^ . TOTTO R 



cos dx = S(N - M) sin + — , ... (23), 



a 71 TT an'' 



R being a quantity which does not become infinite with n. It follows from (23), that the series 

 (22) is in all cases convergent, and its sum for all values of x, critical as well as general, is the 

 limit of the sum of (21). 



It will be observed that if /(.r) is a continuous function the series (22) is at least as convergent 

 as the series 2 — j . This is not the case with the series (3), unless /(O) =/(a) = 0. 



If the constant term and the coefficient of cos in the general term of (22) are given, f{x) 



a 



itself not being known, except by its developement, we may as before find the values of x for 

 which f (r) is discontinuous, and the quantity by which f{x) is suddenly increased as x increases 

 through each critical value. We may also, if we please, clear the series (22) of the slowly con- 

 vergent part corresponding to the discontinuous values of /(.r). 



11. Since the series (3) is convergent, if we have occasion to integrate /(a?) we may, instead 

 of first summing the series and then integrating, first integrate the general term and then sum. 

 More generally, if (p{x) be any function of .t which does not become infinite between the limits 

 r = and x - a, we shall have 



Jr' 2 r" n-rx' , r' . raTr.r 

 I f{x) (p (x) dx = - S / f(x') sin dx . d> {x) sm dx, 

 « -'o a ■'o " 



the superior limit x of the integrals being supposed not to lie beyond the limits and a; and the 

 series at the second side of the above equation will be convergent. In fact, even in the case in 



which f(x) is discontinuous the series will be as convergent as the series 2 — j- • A second inte- 



j;rati<)n would give a series still more rapidly convergent, and so on. Hence, the resulting series 

 may be employed directly, and not merely when regarded as limits of converging series. The 

 Bamc remarks apply in all respects to the series (22) as to the scries (."!). 



12. Hut the series resulting from differentiating (3) or (22) once, twice, or any number of 

 times would not in general be convergent, and could not be employed directly, but only as limits 

 of the convergent series which would be formed by inserting thf factor g' in the general term. 



