338 Uher Beihen cmf der Convergenzgrenze, 



meaning, but it does not thence follow that ^^''X^ has also a 



(iim. o: = ^) 



definite meaning. 



But whenever the sum XJi + U2 + +11^+ of the maxima 



values, which Wi, ii2y assimie upon C, is convergent, the 



expression 



W) 



(lim. 0: = f) 



has always a definite limit which coincides with F(f). It is very easy 

 to derive from this theorem a multitude of others. For instance, the 

 maximum value of - being 



X being positive and q- varying within the circle whose centre is 0, and 

 whose radius is imity 



(1 - o:Y = ^CnP^' (1 - 3;)^ 



7? = 0, ... 30 n = 0, ... 00 



has, for lim. .r = 1, zero as limit whenever ^ Cnjn^ converges. 



Fh') 



If UI + U-2+ + v.n.+ be divergent when x = ^, x — ^) 



will have 00 as limit, if a certain condition be satisfied; viz., the 

 curves described by Un, x varying upon C, should lie in an angle o>, 

 less than tt, whose vertex is 0. This criterion is principally important 

 on account of its connection with a certain theorem, the "type 

 theorem." 



Two sequences 



^h, ^h, Un, 



are said to be of the same " type " if iinl^n, lim. 01 = od , lim. x = f , 

 tends uniformly towards a finite limit p difi"erent from zero. If then 



+ + + Un + be divergent when x = f and the iin 



satisfy the above condition 



+ +^^^ (lim. = 



Vi+ V2+ + Vn 



will tend towards p. 



The second chapter deals -with an application of the type theorem. 

 A power series ^CnOf^ whose coefficients are such that 



n = 0, ... 00 



f0 + fi+ +Cn 



