188 UNIVERSITY OF VIRGINIA PUBLICATIONS 



whose boundary of convergence is the circle about the origin with radius 

 1. The sum of n terms is 



1-2 ~ 1 - P e'» 

 Let p and q be any assigned integers, then any rational point on the bound- 



ary can be represented by « , and any irrational point e'" by a regular 

 sequence of rational points. Let y be an arbitrarily assigned real number 

 and let 



^ -f 1 = mg + r, (« + 1>Q', 0< r <q) (3) 



q and r being integers. 



i-S 



Let 2 proceed, as n becomes infinite, to the point e « on the boundary 

 by the regular sequence of points 



^=(1+^)^'^' 



_ e V rpTT i-(-l)"'P 



*'"n + l"^g'' q{n + l)^ 2(n + l) ' 

 l-(-l)'"p"I 



{n+l)<p=e+\nip + 



(4) 



Then {n + l)^' is always equal to d plus an even multiple of tt, and the 

 limit of the sum of the n terms, or the value of the series at the point on 

 the boundary is 



1-/ • 



(5) 



1— e « 



where ^^y + id is an arbitrarily assigned number, y and 6 being arbi- 

 trary real numbers. Consequently a mode of convergence of z to an arbi- 

 trary point on the boundary can always be so chosen as to make the series 

 take at this point, uniquely as a limit, any arbitrarily chosen number N. 

 If we choose 6 = and 



,^n.inf^. 



