UNIVERSITY OF VIRGINIA PUBLICATIONS 



BULLETIN OF THE PHILOSOPHICAL SOCIETY 



SCIENTIFIC SECTION 



Vol. I, No. 6, pp. 187-199 November, 1911 



AN INVESTIGATION OF THE VALUE OF AN INFINITE SERIES 

 ON THE BOUNDARY OF THE REGION OF CONVERGENCE.* 



BY 



• WILLIAM H. ECHOLS. 



1. The object of this paper is to demonstrate that the function rep- 

 resented by an infinite series of functions of a variable has an essential 

 singularity at each point of the boundary of its region of convergence. 



In order to do this, however, we shall generalize the definition of an 

 infinite series as follows. Let ui (x), 112 {x),Ui (x), . . . , be an infinite 

 sequence of functions of the variable x. We represent by the symbol 



Ml (x) + t«2 (x) + U3 (x) + . . . (1) 



the value of an infinite series of these functions and define this symbol to 

 mean the limit when, n is infinite, of the sum 



Ui [x + <p (n)] + iiq[x + <pin)] + . . . +i02,[x + <p{)i)] (2) 



wherein <p (n) is an arbitrary function of n having the limit zero when n is 

 infinite. At all points of uniform convergence this new definition gives 

 the same results as the definition in current use, at points of non-uniform 

 convergence the results are more general including as special cases the 

 results of the older definition. 



Before proceeding to general series we shall illustrate by a few special 

 examples. 



Example 1 . Consider the geometric series 



1+2+2^+ . . . 



* Presented to the Philosophical Society, October 3, 1911. Read before the 

 Scientific Section at its regular meeting, October 16, 1911. 



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