132 



iJr Kienast, Extensions of 



Finally suppose that a^lbn tends to tlie lint it I luhen n tends to 

 infinity. Then 



lim (S a^afjt b^,/f) = I (o), 



tuhen X approaches 1 along any path inside D. 



This theorem is due to Pringsheim*. It is to be supposed 

 throughout this paper that, when x tends to 1, its approach to 1 

 is along some path inside B. 



Theorem 7. If the radius of convergence of F {x) = '!E a^x" 



is r, then 



lim anX'"' =0 (\x\< r). 



If the radius of convergence of Q(x) = Xa^x" is unity, it will 



I 

 remain unchanged if Q(x) be transformed in any of the following 

 ways : 



(i) by suppressing a limited number of terms, 



(ii) by multiplying by ./;°", a being an integer, 



(iii) by multiplying by ^ _ ^ , = ^ «'\ 



(iv) by integrating term by term, 



(v) by differentiating a limited number of times. 



Using in succession one or other of these operations, there 

 result the following power-series, all with radius of convergence 

 unity : 



xF' (x) — S Ka^x", 



\(x)= —A^F'{x)] = lrl'>x, 



X — X I 



F,(xy= ^ 



^ 1 ~l '^ ) 



- Pj (x) dx \ =X r,^ x", 



1 — X 



P.^ (x) dx 



Pg (x) dx 



= ir^^x'^\ 





* 4Qta Mathematica, vol. 28, p. 7. 



