Taylor and Maclaurin Series f(x) = x sin(x) Compute f(0) = 0 f'(x) sin(x) +x cos(x) f'(0) = 0 2 cos(x) -x sin(x) f(0) = 2 f(x) = 3 sin(x) = x cos(x) f(0) <=0 f)(x) = -4 cos(x) +x sin(x) f(u)(0) = f)(x) = 5 sin(x) + x cos(x) f() (0) = 0 We see that for the odd terms f(2+1)(0) = -k cos (0) and we also see that for the even derivatives f(2) (0) - k cos (0) Hence the Taylor series for f centered at 0 is given by 2k f(x) = (-1) 2kx2k (2k)! = x sin(x) for k21 except for k = 0.