190 UNIVEESITT OF VIRGINLl PUBLICATIONS 



Let z{ = )0 by the sequence s = $"/«, j" arbitrary. The value of the 

 series then is sVCf + 1) or ((— co, + od)). 



Exam le ' V 'A n{n+lV-l] ^ z _ (w + D^ 



X mp e ^. 2_, (1+^222) [1+ („ +1)2^2] 1 ^ .2 1 + (71 + l)=/ 



When 2( = )0 by the sequence f/(?2 + 1) the value of the series is ^/(l + l~) ■ 

 Example 5. ^ + ^ ( ^^^^ - ~^^^) = ^^^^ 

 The sequence s = 1 + 1/(2?! — l) gives S^ = es 



Example 6. > , „, — = ^ r- 



-^[l+(l+z)"-'][l + (l+2)"] l + (l + 3)" 



z{ = )0 in the manner ? =i^n gives .S„ =(1 — ef)/(H-ef). 



1 ^ s"-/'+' 1 ■ 

 Example i . + > -r = 



1+c ^fi+j")ri+.j"+n 1+3" 



Sco =1 for l^] <1, and .§„ = for \z\>l. If z goes to the boundary in 

 the same way as prescribed in example 1 the value of the series at a definite 

 point on the boundary is 1/(1 + eO- If 2 is confined to real values and 

 goes to the point 1 by the respective sequences 



\ z = ± — 



n 



the value of the series at 1 is (l + e*'')""' or the totahty ((0,1)). The 

 series has there a shear represented by a vertical segment joinmg the value 

 of the series for \z\ < 1 to its value for \z\ > 1. 



Examples. A similar example, due to Tannery, having infinitely 

 many poles at roots of unity on the boundary is 



1 z z- z" , 



1 — i z — 1 <r—\ z^—\ 



The sum of its first n terms is — (z-""' — 1)~''. The value of the series 

 is 1 or according as \z\ is less or greater than 1. Let 2 converge to a 

 point on the boundary as in example 1, where in (4) n + 1 is replaced \>y 

 2"~\ The value of the series at this point is then l/(e^— 1). 



