INTO ARITHMETICAL EXPRESSIONS FOR INFINITE SERIES. 191 



The result may be easily verified. The given series may be thrown 

 successively into the forms 



1. U-<1) (1 -;r^)f_^ +_£!_+ _£L_. \ 



or {l-x'){x + a^ + x'+ ...} , that is, w. 



2. (x>l) (a-=-l)/-£_ + -^+_^_+ 1 



M^ - 1 a;' - 1 ^ a;"= — 1 ^ J ' 



(a;«_l)/l + i+ 1 , I 

 '[af^ of x' ^ •■■]' 



1 



IS, -. 



The second example in my former paper was the series 

 £ ax a^<(; 



(l+a;){l+ax) (1 + ax) (1 + efx) "^ (1 + a'x) (1 + a'x) + " 



^■^ " (1 +x){l + ax)' ^"^ = ^' "^ = «^' "°°* = «"^-- 

 The equation of the series is 



X 



tpx = 



(l+x)(l+«x) + '^<'^^)' 



a particular solution of which is «a; = I. 



(«-l)(x-l)' 



and a particular solution of (px = cp (ax) is x = C = vx, 



fxa" X 1 1 



nx vx = 



va"^x {a-l)(x + l) (a- l){a^x + 1)' 



when a > 1 = 



when « < 1 = — 1 £. 



(«- l)(a;+l)' 



1 

 (« - 1) (a; + 1) ~ «- 1 ^ (1 -«)(a; + 1) 



This result may also be verified; for the original series developed 

 term by term in powers of - and -, and corresponding series of 

 powers of - collected gives 



