OO Cn (2 - 3)n Write the Taylor series for f (x ) = e about x = 3 as n=0 Find the first five coefficients. CO= S1= C2= C3= C4=By recognizing each series below as a Taylor series evaluated at a particular value of x, find the sum of each convergent series. A. 1 + 3 + ( 3 ) 2 + ( 3 ) 3 + ( 3 ) + + .. . + ( 2 )n+...= B. 1 -. 52 54 56 4! 6! + ... + (-1)252n (2n) ! +... =6 + 3x Let f(a) = Compute f(ac) = f(1) = f' (ac) f' (1) = f" (ac) = f" (1) = fill (ac ) = f'' (1) f ( iv) ( z ) = f (iv) (1) = f ( 0 ) ( 20 ) = f ( 0) (1 ) = We see that the first term does not fit a pattern, but we also see that f (k) (1) = for k > 1. Hence we see that the Taylor series for f centered at 1 is given by f(20) = 9+ (x - 1)k. k=Find the Taylor polynomial of degree 3 for sin(z), for near 0: Py(z) = Approximate sin(z) with P3(z) to simplify the ratio: sin(z) _ i Using this, conclude the limit: z0 z cos(ac) - 1 + 2 Evaluate: lim x->0 9x4 Hint: Use an appropriate Maclaurin series from Table in secton 11.10 in the text.Consider f(x) = cos() = (-1)~ x2n n= 0 (2n)! Suppose T2 (a), the Taylor polynomial of degree 2 about a = 0, is used to approximate f(x). If |a|