EXPANSION OF FUNCTIONS. 93 



Similar remarks apply to the expansions in the next two 

 Articles. 



121. Expand ^ tin ~ 1 ' in powers of x. 

 Put e a ~ lx = y (1), 



flion *L /jOgin~'x ff)\ 



WlCil 7 K ., , V-/ 



flip / I 1 Sf>* I V / 



t(iO , . 1 Ji, I 



+ 



da? 1-x* 



\- "i 



therefore (l x 9 }-~, a:-^ = a 9 y (4). 



' dx dx 



Assume y = A^ + A^x + A^a? + A^x 5 + . . . + A n x n + . . . (5) ; 

 dif 



f npi'pfViTP * - A Ji- A fp A -L fn A /y n "~^ I -, 



LllCl t/lVlC 7 "^1 T^ ^-iioW T^ * T^ It'^l.ftLJU ^^ m 



ti (ii 



Substitute these values of y, -/- , and ,,, in (4), then equate 



u3o aoo 



the coefficients of like powers of x on both sides, and we 

 obtain 





Equation (6) will enable us to determine A 2 , A 3 , A t , ... as 

 soon as we know A and A r 



But A is the value of y or e at ^~ lx when x = 0, and 



^4. is the value of -J^ or e? 1 *' 1 *-^ - ^ , when x : 

 dx ,J(l x) 



therefore A = I, and A^ = a. 

 Hence, by (6), 



A -^-A -- 



^~^~ 



