224 



* to 



Theorem A: The ctnulitlons /Im. 2 ii„.)-"=s and /i7n.n[A,f —A,!^li]=0 



.1 — *1 1 n=co 



are each necessanj for the e.ristence of: 



Urn. An —s, 



and taken together the;/ are snffcient. 



Indeed (lie necessity of the condition //ni. ^ a,„.v" = s follows 



from Holder's theorem mentioned above, and the necessity of 



//m.n[^jir'— /ll'-i]=--0 is seen by writing it //m. [.-ilf' ''— jL'1, ] =0 



and by observing that lim. An~ ' = s implies Urn. A„' ^ x. 



The following particular case of tins theorem has been proved 

 by Tadbür ') : 



00 



Theorem B. The conditions Jim. 2 a».*;" ^= s and 



.r— ^1 1 



1 



l/m. - (//., -4- la, -|- . . . . + na„) = 



n=QO ?i 



are both necessary for the convergence of 2 an, and taken together 



the;/ are sufficient. 



This may be seen by substituting /? = 1 in theorem A, for: 



4"" - s 



r\n — ON 

 " [A„ — An^\ \^= An — An-\ = «n ; = 



n— 1 

 1 1 



Ml n — 1 



1 



= [(«, + «, 4 • ■ . + "") + («,+••• + O'n) + ■•• + fln] 



n — i 



1 



= r [(«—!) f'n + (" — 2) a„_i -j- . . . + a,] 



n — 1 



1 " 



= r -2'(p-l)a„ , 



n — I 1 



and we may infer the equivalence of the conditions 



1 1 



hm. \a^-\-2a,-\- ...-{-{n-l)a.,?\^=0 i\.nd Inn.- {a^-\-2a^-\- ...-\-na,)z^O 



n=oo ?i -* M=roo /i 



from the equations: 



U {x)=:za.^ X -\- a, j;' -|- . . . ; V{x) = a, a; -|- «, ^' + . • ■! t/ (.r) = a, x -)~ a; F («) 



1) Bromwich, op. cit., p. 251. 



