424 Mr Hill, On the series for e x , &c. [May 10, 



Putting now n = 1 on both sides of the identity of Art. 2, it 

 follows that 



(x x 2 , x 3 \ 1 (x x* x 3 \ 2 

 1 + (l+2 + 3 + -) + 2!(l + 2 + 3 + -) + - 

 = 1 + x + x 2 +x 3 + ..., 

 which is a Geometrical Progression to an infinite number of terms. 

 Since x < 1, its value is ; 



1 — x 



1 — X 



%AJ tAs %Aj 



■"• j+ 2" + g- + --- = -log e (i-*'); 



rp rp /p° 



•'• log« (1 + «) = j - 1 "2 + 3 ~ ••• if x < !• 



Art. 4. The Binomial Theorem. 

 Taking x < 1, 

 (l-cc)~ n = e- nl0 ^< 1 -^ 



(cc oft cc^ \ 

 1+2 + 3 +..) when x< 1 by Art _ 3 



= 1 + ' ! (l + 2 + 3 + -J 



n 2 [x x 2 x 3 \ 2 . 



+ 2Hl + 2 + 3 + "\) + - b y Art -!' 



_, w (n + 1) „ 



= 1 + nx + v 2 ^ ' x 2 + ... 



n(n+l)...(n + r-l) r , . , 



H r — — - x r + ... by Art. 2. 



Changing the sign of x, and putting n = — m, 



/-i \m i m(m— 1) „ 



(1 + x) m = 1 + w« + — i— - * « 2 + . . . 



m(m— 1) ... (m — r + 1) ,. . 



4 r i #+..., when x < 1, 



r ! 



which is the Binomial Theorem. 



