Question

2. The Taylor series of the function f(x) = - iſ about x = 0 is given by (x − 2)(x2 – 1) 3 15 15 2. 63 4 F=3+ = x + x2 + x + x4 + ... (x − 2)(x2 - 1) 8 16 6 (a) (6 marks) Use the above Taylor series for f(x) = . T and Calcu- (x − 2)(x2 – 1) lus to find the Taylor series about x = 0 for g(x) = ln (1- F " (x-1) 3 T : not use Taylor's theorem or trying to solve for the Cn coefficients directly. You may assume absolute convergence. Hint: While the functions f and g may seem unrelated, you only know a few Calculus techniques for finding Taylor series. Apply one of these methods to f to produce g. |(x+1)(x-2)2). You may (b) (2 marks) What is the interval of convergence (that is, the set of x values for which the series is valid) for the Taylor series of g(x) that you found in (a)?

Answer 1

See that for 20 (21) = 1 + 2 2 2 2 2 2 2 Assuming uniform convergence of 3 3 žu + $#**FR + we can integrate term by term : (1-2)(x²-1) NON of the Season 34 +23+27+. 2+1 luly+11+2 lu19-21-2242 - 3 lufyll n a 3y + 2 y + 5 3 tone 7 lujo M-0332) = 24u2 + 3y + å yttäysten e. g(x) = 211 2+ 3x + 2 + s tra luf 9+1) (9-232 + (6) We know interval of convergence for Taylor series of Infatto lulatll is alkaen so interval of convergence for Taylor series of -dela-ll is also al call and that of lu12-21 = lu2tlul 7-11 is <3 <1 iic.-2<x<2 & Toning the common interval - 1<x<1 we get the interval of convergence for Taylor series of g(x)