THE SUMS OF PERIODIC SERIES. 553 



however in which /(.r) does not change sign in passing through eo , and in some cases in which 

 it does change sign, /(a + e) + /(a - e) becomes infinite when e vanishes. 

 In such cases put for shortness 



/(« + — D+/(«--D = ^(D> 



and let the numerical values of the integral / — ^-^ df taken from to - , from — to - — ... 



or which is the same those of - -^ d^ taken from to tt, from tt to Stt ... be denoted by 



/,, /a ... Then evidently I^> Ii> I^ ... Also, if ^ be sufficiently small, F(^) will decrease from 

 ^ = to ^ = ^, if we suppose, as we may, F{^) to be positive. Hence the integral (42), which 

 is equal to 



-j/,F(e,)-/.F(Q+/3F(^3)-...u (4.<i), 



TT 



where P,, ^2 ... are quantities lying between and _ , — and — ... is greater than 



V V V 



IT 



if we neglect the incomplete pair of terms which may occur at the end of the series (43), and 

 which need not be considered, since they vanish when 1/ = cc . Hence, the integral (42) is 



a fortiori > — (Z, — 7^) F(^^). But ^1 vanishes and /'(^i) becomes infinite when v becomes infinite ; 

 7r 



and therefore for the particular value x — a the sum of the first n terms of the series (3) increases 



indefinitely with n. 



If a coincides with one of the extreme values and a of x, the sum of the series (3) vanishes 

 for a? = a. This comes under the formula given above if we consider the sum of the series for 

 values of r lying beyond the limits and a. The same proof as that given in the present and last 

 article will evidently apply if /(r) become infinite for several values of ,v, or if the series considered 

 be (22) instead of (3). In this case, the sum of the series becomes infinite for x = a when 

 a = or = a. 



2.3. Hence it appears that/(.r) may be expanded in a series of the form (3) or (22), provided 



only ff(v) dx be continuous. It should be observed however that functions like I sin - J , which 



become infinite or discontinuous an infinite number of times within the limits of the variable Hithiii 

 which they are considered, have been excluded from the previous reasoning. 



Hence, we may employ the formuloe such as (26), (35), &c., to obtain the direct developenieiit 

 of /*'(.r), without enquiring whether it becomes infinite or not within the limits of the variable for 

 which it is considered. All that is necessary is that /(.r) and its derivatives up to the (m - 1)"' 

 inclusive should not be infinite within those limits, although they may be discontinuous. 



24. In obtaining the formulae of Arts. 7 and 13, and generally the formula; which apply to 

 the case in which A„ or B„ is given, and f{x) is unknown, it has hitherto been supposed that we 

 knew a priori that f{x) was a function of the class proposed in Art. 1 for consideration, or at 

 least of that class with the extension mentioned in the preceding article. Suppose now that we 

 have simply presented to us the series (3) or (22), namely 



11 TVX TlTT X 



'^A„ sni - or H„ + '^B,, cos • , 

 a a 



