Question

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that R(x) - 0.] f(x) = x4 – 3x2 + 1, - 2 (2)(x - 2)" - 5 + 20(x - 2) + 21(x - 2)2 + 8(x - 2)2 + (x - 2)4 x - 2)" = 5 + 20(x - 2) - 8(x - 2)2 + 21(x - 2) - (x - 2) į myl2)(x - 2) - 5 + 2063 ſ l2)(x - 2)2 = (x - 2)" - - 5 - 20(x - 2) + 21(x - 2)2 + 8(x - 2)2 + (x - 2) . 2)(x - 2)" - -5 + 20(x - 2) + 21(x - 2)2 - 8(x - 2)3 + (x - 2) x - 2)" - 5+ 20(x - 2) + 8(x - 2)2 + 21(x - 2)3 + (x - 2) Find the associated radius of convergence R. R-

Answer 1

2) = 248 f(x) = x4_3x² + f(2)= 24_3(2²) +1=5 f'(x) = 4x² - 6x f'(2) = 4 (23)_6(2)= 20 f (x) = 120c²_6 of "(2)= 12 (22)_6= 42 fir (2)=24(2) = 48 . f(v) (21) = 24 if(iv) (2) = 24 .. f(K)(x)=0 akys : f(K) (2)=0, K7,5 Taylor series of frow about s=2 is fræ) = f(2) + fle) (A-2) + f * 2) (=232 + + Flv (2) (2-2)4+---- 11 - 84-34?+1= 5+ 206x-2) + 42 (06-232 + gas (x-pja + 2 666-234+0+.... > 24–3x2+ 25 + 20 (6-2) +21 (8-10278 (0-2)3 +(x-2)4 ophon Since Taylor series of given f is a finite Series . It converge for all at Coorool .. o R= coll