206 Mo' Rohb, Discussion of a difference equation relating to 



Multiplying corresponding sides we get 



2.4.6.8...2W 2w + l 



< — ^ — 



1.3.5.7 ...{2m -I) 

 if m > 2. Or 



4.G.8...27?i „ 1 



<-' 'in- 



1.3. 5. ..(2m -3) 4 



Thus finally 



4.6.8... 2m 



< m-. 



1.3.5...(2w-3) 

 For m = 2 this inequality becomes an equality. 



It follows that 6.J, bs, h^, h, ... 



are respectively greater than 



Oo, a/, a/, a^, ... 



where «/= 1, 



1 



1 



a; = 



^' = 2(F:ri)T^(^«^" + 6">^')' 



and 7 is less than unity. 



A fortiori 



h, h, h^, ... 



are respectively greater than 



0-2, a-i, «4, ••• 



and therefore for values of 



the series 



y = 1 — a- — a..af — asOf — a^a;-* — . . . 



converges more rapidly than the series 



1 , 1.3 . 1.3.5 , 



v = l-a:-^^^v--^ar--j^cc^-.... 



If we include terms up to that involving a--'" the remainder for 

 this latter series is known to be numerically less than 



1.3.5 ...{2m-l) x'^+' 

 m + l\ 1 -2a;' 



