418 Mr Hill, On the series for [May 10, 



•■■ / H =/(- n ) =J^ = JJJ^n , by the previous case 

 = [/(i)]"™, by the definition of a negative index 



= [/(!)]*• 

 Hence whatever a? may be 



/(*)=[/ (I)]*- 

 But /(l)=l+l + 9-f + q-j+--- which is usually denoted by 



the symbol e, 



.-.f{x)=e*, 



x 2 -x* x r , 



•• ^=1+^ + 21 + 3;+...+-,+ ... 



whatever x may be. 



The series is convergent so long as x is finite. 



Art. 2. It is required to establish the identity of the series 



, r z? n r (x x 2 x 8 , 



1+2 -. T +^+ ... + - + ••• 



r=1 rl \1 2 s 



, , , J '= co w(w+l)...(w + r-l) , 



and 1+2 — — -x r ; 



r=i H 



the summations being effected for all positive integral values of r. 

 The first series is 1 + n f - +-%+ ••• -I \- ■■■) 



n 2 (x x 2 x r \ 2 



+ 2l(T + 2 + - + 7 + -) 

 + 



n r (x x 2 x r 

 + r-l(l + 2 + - + 7 + 

 + 



Let it be written for brevity c + c t x + c 2 x 2 + . . . + c r x r + ... 



Then c r = n . - 

 r 



2< 1 1 , fla 1 + 2a 2 +...+ra r =r, 



+ 7l2 'a 1 !a 2 !...a r !-1^2« 2 ...r-' Wnere [ « x + « 2 +...+ « r = 2 

 + n .*_i I where {l«t + 2«,+ ...+ra r =r f 



+ 



rV 1 1 , fla 1 + 2x 2 +...+ra r = r, 



+ 7lZ a 1 !a 2 !...a ! ' 1«>2«« ... r" ' ° | « x + a 2 +...+ « r = r. 



