82 Mr Kienast, Proof of equivalence of different mean values 



Substituting (14) in (11) we find, on adding a term which is zero, 



„(!)_ 1 r^2^ ,(1) , ^sf^ .(1) p Ai)x , 



V ih_h±i] c t^^^ 



Since 



1 VCa Ca+1 



+ 



^n+1 ^n ,(1) 



-6. 



^n+i 



G,= i^~\C\-C\_,) + 



2 Ca 



= 5. 



we can write 



1 _ '"'+1 zl 







a, 





- i "+ 



Ca Ca+i/ I 



2 ( ^ — ^^^ 

 ,Ca Ca+1 



C, 



.(15). 



Now there may be distinguished two possibiHties : 

 Theorem 7. If 



VCa Ca+t/ ^ ^ 



w /A 



Ca fA+] 



C. 



< K (fixed), 



0„ 6,1-1-1 



lim — = 1 



?i-*.Qo '^n+1 -'-'71 



and if ^JJ' a'pproaches a finite limit {or oscillates between finite 

 limits), then s^^^ approaches the same limit (or oscillates between the 

 savne limits). 



Theorem S. If 



" /&A &A+A p (1) 

 ■ i "-'a ''A 



Hm 



V /"A 



3 VCa 



= lim i^ 



^()a_6a+i\ ^ 



2 VCa Ca-|-i/ 

 b G 

 -~~^<K (fixed), 



and ift^^^^ approaches a limit, then s^^ approaches the same limit 



This is a known theorem *. 



The second relation results by substituting from (13) in (12) 

 and proceeding in the same way.' The same formula is arrived at 

 by interchauging in (15) b^ and c^, B^ and (7,^, ^'^ and s^^^ . From 

 it we infer two theorems analogous to (7) and (8).' 



* Bromwich, Infinite Series, p. 386, Theorem V. 



