a Necessary and Suficiejit Condition /or Uniform Convergence 193 



Hence the sum of the series, when 6 = 7r/2X,i, can in no case 

 tend to zero, as n tends to infinity, unless X,i (a,i — a,n) -^ 0. 



If X)i (an — a,n) -* 0, then, given any positive number e, we can 

 find V such that X,i (a„ -«,„)< e for n^v. Denote ni by (n, 1) 

 and let (/i, 2) be the integer formed from (n, 1) in the same way 

 that {ii, 1) is formed from n, and so on. Then 



((n - «n, 1 < e/\n , (/„, i - ttn, o < e/X^, i , , 



for n ^ V, and by addition 



Un < € (1/X,„ + l/Xn,i + ... + lAn,i>) + (hi,p. 



Now X,i_i > 2\,i,, \,i,^2 > 2X„_i, and so on, so that «„ < 2e/\„ + cin,p. 

 Also when % is fixed we can choose p so that an^p < e/Xn, and we 

 shall have therefore 



Xnttu < Be (n ^ v). 



Hence XnCin^O is a necessary condition that the sum of the 

 series should be continuous at ^ = 0, and a fortiori that it should 

 be continuous throughout any interval which includes ^ = 0. 



3. To show that this condition is sufficient for uniform con- 

 vergence in any interval, and d fortiori for continuity at any point, 

 it is sufficient to show that 



I Un+i+ ... +Up\ <AM, 



for all values of 6, where A is some fixed number and M is the 

 greatest value of Xt-a^ for r^n + 1. 



Since the value of the series is changed in sign only by changing 

 the sign of 6, it is sufficient to consider positive values of 6 only. 

 By Abel's lemma 



I Un+i + ... +Up\< 2an+i/d < 2Xn+ian+i/7r, 



if ^ ^ tt/Xji+j. If ^ ^ T^/Xp, every term of Un+i + . . . + Up is positive ; 

 and, if u,. is one of these terms, 



Ur ^ M (cos Xr-iO — COS X,.^)/X,.^ 



^ 2if sin l(Xr - X,_i) 6 sin h{X, + X,_,) O/X^d < MO {Xr - X,_i), 

 so that Un+\ + . . . + Up < MOXp < ttM. 



If irjXp < 6 < 7r/X„i+i, let Tr/Xq+i < 6 ^ 7r/Xg, and divide 



lln+i + ... ^Up up into Un+i + . . . Uq and Uq+i + ... +Up. 



Then | Un+i + ... + Uq\< ttM, and 



I Uq+i + ... +Up\< 2aq+^l0 < 2aq+iXq^,/7r < 2ilf/7r. 



Therefore j Un+i + ... -\- tip\ < {ir + 2/7r) M. 



U~2 



