OPERATIVE DIVISION 235 



H. Given the series for e x , find by operative division the 

 series for log (1 + y) ; and vice versa. 



Put e x — 1 = y, so that x = log (1 + y). Then also by 

 operative division, 



"*+£ + . 



L 2 ! 3 ! J 23 



and vice versa. 



I. Treat similarly the series for sin x and sin -1 *, and other 

 trigonometrical functions. 



J. 4>(x) . 6* = *(*), 



where <£(#) and yjr(x) are rational integral functions or can be 

 reduced to such. Expand e x , collect coefficients of the same 

 powers of x, and invert. Show how to render, if possible, the 

 series convergent by suitable transformations. 



K. o = fix) ; 



if f(x) can be expanded in an ascending series. Expand 



f(p + 2), invert by the method of (4) above, and select p so 



as to render, if possible, the series convergent. 



L. Given the binomial theorem for integral exponents, 



prove it by means of operative division for fractional exponents. 



Let (i + x) n = y, where n is a positive integer; so that 

 1 1 



x = y Vi — i = {i-j-(j/— 1)}"— 1. Then 



. n(n— 1) , , 

 1 + nx + — — j — L x l + . . . = y ', 



, n — 1 . , n — 1 n — 2 , , y — 1 



x-\ x t -\ #'+ . . . = •. 



223 n 



Inverting as usual, 



x _y-i w-i/y-i\' f a /w-iV' n-in-2 )/y-i\> 

 n 2 \ n / [\2/ 2 3 J\ » / 



= L (y _ I)+ i(i_ I \b^ + i(i_ I yi_ 2 )(z = j)! +etc . 



n ^ ' n\n / 2! n\n }\n J 3 1 



1 

 = {1 + (y — 1)}" — 1, by the binomial theorem. 



M. If <f> = — co 8 + do 3 — etc., compare the coefficients 



of the operative quotient of = with those of the algebraic 



quotient of -, and show that they are composed of precisely the 



