T A Y 



Ex. 1. To expand (.r 4. h) m . 



Let u = a? m and u' = (x 4- h] . By differentiation we have -3 



m-i rf w v m- 2 efe M , w-* 



w .1- j *-- = TO.. (?-!)* ; -T- w . (OT - 1) . (m - 2 x &e. 

 ' d j-* ^'3 



Hence, by Taylor's theorem, v,' a- + mir ^4- w ~^~~ * v " ^ z 

 + &c. 



i\r. 2. To expand ( ^- *}"* by Ma^-laurin's theorem. 



Let u - ( + *) W ; .'. -^ = m. (a + jf^ 1 ; ^ ^ m. (m - l.J 

 (a 4- .r) w - 2 c. &c. Now let * = 0; then (a) = a m ^ ( ^i) 

 = m a m " 1 j ( ^) =c w, (m - 1.) a m ' 9 &c. ft 



= (a + z) m - a + W " ! * + ~ ! m " 2 4- &e. 



jr. 3. To expand a' in a series. 

 Let u = a T , then - = ft M (ft 

 ' 0, then u 1 ; .*., by Madaurin's theorem, 



Let u = a T , then - = ft M (ft = h. 1. a), j-~ ft* w, &c, Now let 



Ex. 4. To expand log. (x + ft). 

 Lot u I x, and W = / (* + A) ; .*. 



du _ m dzu _ w <^3 M _ 2w _ 

 ~dx = d^2 P'JI^"" 1^~ 

 .*. by Taylor's theorem, 



x 7i A2 , ^3 A* 



'= w + in ( 7 - ^ + ^ - 4^7 + A 

 Cor. If x = 1 , we have 



7?or. 5. Expand sin. x in a series. 



I>et w = sin. .T. Take the successive differentials of sin. r, and find 

 293 R2 



