a Necessary and Sufficient Condition for Uniform Convergence 195 



It follows also that 



i -& I 



I Z sin (X^+i - \,)6 sin X,^^ < 2 + Se^ < 3, 



|w + l I 



for n ^ V. Now 



(X„+i - Xn) 6 cosec (X.,1+1 - X„) ^ - 1 

 decreases steadily to zero, and is less than 



Therefore 



2 6{\n+i - \i) sin Xu^ — 2 sin (X„+i — X„)^ sin XnO 



n+l jj+1 



Hence 



< e'6' 



p 



% {(cos X,i^ - cos \n+i6)ld - (Xn+i - ^/i) sin \nd} 



n+l 



< 3e + e^O < 4e (n ^ i^). 



Hence the series 



2o„ {(cos Xn^ - cos \n+id)l6 - (X„+i - X„) sin X„^} 



is uniformly convergent throughout any interval, and hence the 

 result enunciated follows. 



5. If instead of a sequence (X„) we have a function X (x) such 

 that, as iT-^oo, X{a;) increases steadily to infinity and \'{a;) de- 

 creases steadily to zero, then Xn+i — ^n decreases steadily to zero. 

 The series 2 (X'„ — X^+i + X,i), where \'n denotes the value of X' (x) 

 when x = n, is convergent and is moreover absolutely convergent, 

 since X',^ — \n+i + Xn is positive. Hence, by Weierstrass' M test*, 

 the series Sa,i (X'„ — X,i+i + X,i) sin X„^ is uniformly convergent 

 throughout every interval. It follows then that ajj,X„ ^ is the 

 necessary and sufficient condition that the series Sa^X'„ sinX,j^ 

 should be continuous at every point and uniformly convergent 

 throughout every interval. 



In particular the series 2a,i?i'~^ sin (n'^), where t is any real 

 number not exceeding 1, is continuous at every point and uniformly 

 convergent throughout every interval if n^a^-^O, this condition 

 being necessary as well as sufficient. 



* Bi'omwich, Infinite series, p. 113. 



