Question

Find the Taylor series for f centered at 4 if f(n) (4) = (-1)" n! 3" (n + 1) What is the radius of convergence of the Taylor series?

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. (Assume that f has a power series expansion

f(x) = cos x

Answer 1

so, by Cauchy - Hadamars test the radious of convergence of the Taylor series an (21-407 is 3. #1. Let f(1) be a function and given that (-15hn! 3n(n+1) ". The Taylor series expansion of the function of centered at 4 is f (21) = 3 h 14) (x-435 no n! f(1) = (-15h (x-472 no 3h(nt) (-uh Let ana n7o. 3h(n+1) Now, find the radious of convergence of the power series ² an (x-4). no e ant) (-ight 3h (nt) ght) (nt2) Toh (1+1) (1+2/n) (n+1) 3 (2+2) anti an hd htd (1+) 1+) - (140) = no

txn h=0 n! fRn) (2) = (-1) coss, f (2n) (0) = (-1) Cojo #2. let f(x) = cosa we find the Maclaurin series of f(x) by definition : The Mactaurinseries of f(x) w f(x) = f (0) + 40 + 1"() + .. man - 29 Aho f(0) = coso = 1 f (x) =- sink f'(o)=0. f"(x) = -cosa f" (o)=-1 f(x) = sinn f(0) = 0 ftyrn) = & con f 14 (0) = 1 = (-1) pknol) (x) = (-1)" sink, fent) (0) = (-1)" sino D + f&h! (0) 224-1 n=1 (2h-1)! f(0) = J Rh) (0) 2124 (2h)! (-1)h x 27 (2n)! & -rxan (2h)! h=0 17