SECT. III.] LIMITS OF THE REMAINDER. 151 



and the limits between which the remainder is included must 

 be found. 



We will apply this remark to the equation 



1 1 1 



y = cos x - cos 3x + - cos ox ^ cos tx ... 

 3 o 7 



~ 



2m - 3 2m - 1 



The number of terms is even and is represented by m ; from it 



Zdy sin Zmx , . ,, 



is derived the equation = - , whence we may infer the 



CtJO COS 00 



value of y, by integration by parts. Now the integral fuvdx 



may be resolved into a series composed of as many terms as 



may be desired, u and v being functions of x. We may write, for 

 example, 



I uvdx = c -f u I vdx =- \dx Ivdx + -j ., Idx I dxlvdx 

 J J dxj j dx J J J 



an equation which is verified by differentiation. 



Denoting sin 2mx by v and sec x by u, it will be found that 



2// = c -T sec x cos 2mx +^r- 9 SQC X sin 2??^ + ^ o sec&quot;o; cos 2 



K 7 sec&quot; x \ 



*-&?-** *)&amp;lt; 



186. It is required now to ascertain the limits between which 

 the integral -^3 , I [d(sQc&quot;x) cos 2nix] which completes the series 



is included. To form this integral an infinity of values must 

 be given to the arc x, from 0, the limit at which the integral 

 begins, up to oc, which is the final value of the arc ; for each one 

 of these values of x the value of the differential d (sec&quot; x) must 

 be determined, and that of the factor cos 2mx, and all the partial 

 products must be added : now the variable factor cos 2mx is 

 necessarily a positive or negative fraction; consequently the 

 integral is composed of the sum of the variable values of the 

 differential fZ(scc&quot;.r), multiplied respectively by these fractions. 



