217 



primo: it is possible whatever he 6^0 and 5<C^x„, to calculate a 

 iiiuuber i/(6,5) so that whatever be y, between ij and ?/„, we can 

 calculate a number .i\ between 5 and ,(„ which satisfies the condition 



l/(*.)-.9(.V,)l<f, 



secnndo: it is possible whatever be e ^ aad »/<C?/i, ^o calcniate a 

 nnniber §(s,it) so that whatever be a\ between g and a;, we can 

 calculate a number //, between ?j and j/„ which satisfies the condition 



00 



Theorem 1. If hm. na,, = 0, i:a„x" osciUntes for .c — *• 1 oèo/zi <Ae 



AY/?»*; region as 2!ai for n —^ oa. 

 



Proof: We have l)y a well-known theoiem ') that lim.ti., =: ?« implies 

 /////. — £ ?(,- ^ u. Hence, since I/in. na,, = implies /im. n \a„\ =0, 



V=C0 I' n=0O »^r CO 



1 V— 1 



lim. — ^ n I a„ I =; 0. 

 »=« 1' u 



Therefore, whatever be f > 0, we can calculate an integer (n so 

 that if r > ft : 



»» I «" l< Y ••••••.(!) 



1 "-1 E 



- :s « I «, |< - (2) 



1' 'J 



Now, if 0<^.(:<^1, we have: 



V — 1 ^ V — 1 a -1 « 



I ^ a„ — ^ fl„ «" I < I ^ a„ — ^ a„ .c" I + \S a„ x" \ 

 00 V 



V— 1 0° 

 < 2 a„ (I— a") \ + \2a„tc" \ (3) 



V 



V— 1 V— 1 V— 1 



I 2 «„ (1— .1;") I < ^ I a„ I . (1 -;r") < (I—,:) . ^ „ | a„ I . . (4) 

 00 ' 



1 



Substitution of .*v^l in (4) gives: 



X' 



M Bromwich, Theory of Infinite Series, p. 383. 



