Turn over 19 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 9 marks) 20 6. 2 ( ) f ( ) ( ) f ( ) f ( ) ( )f ( ) f ( ) ( ! ( f ) ) 2! =     − −   = + − + + + + ′ ′′     … … r r The Taylor series expansion of x about x a is given by x a x a x a x a a a a r Given that y = (1 + ln x) 2 x > 0 (a) show that d d 2 2 2 2 y x x x = − ln (4) (b) Hence find d d 3 3 y x (2) (c) Determine the Taylor series expansion about x = 1 of (1 + ln x) 2 in ascending powers of  (x – 1), up to and including the term in (x – 1) 3 Give each coefficient in simplest form. (3) (d) Use this series expansion to evaluate lim ( ln ) ( ) x x x x → − − + − 1 2 3 2 1 1 1 explaining your reasoning clearly. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________