4. (a) (5 pts) Manipulate the known Maclaurin series representation of 1/(1-x) to find Taylor series, centered at a = -2, which equal the given function f(x) = tan-1(x + 2) on some open interval containing a, and give the largest open interval on which you're certain that the equality holds. "Give" the series either by writing the general series using summation notation, or by simply writing out the first 5 non-zero terms and then + .... J ( x ) SER (-2) S (3 ( 2) / ED /3 + + ( x + 2 ) + 29 ( x + 2 ) too = tan ( * + 2 ) 0 (1- x ) ' -3 1 ( 1 - * ) 2 3 + 9 ( * + 2 ) + 2 ( * + 2 ) ? + 2 ( * + 2 ) 3 + qu3 ( + 2) " + 729 ( 2 +2 ) = tail ( * ) -2 ( 1- x ) 3 2 2 27 29 3 6(1-x)4 6 -2 81 - 29 81 4 - 24 (1-x)5 -24 - -8 -243 81 243 5 120 ( 1 - x ) - 6 120 40 929 243 729 (b) (3 pts) Use the first three non-zero terms of your Taylor series from part (a) to estimate the value of f(x) at x = -2.1, and compare with the answer from a calculator. Aldance 13 + ( - 0.1 ) + 29 ( - 0.1 ) 2 = 01 3 2 25 9 25 926 76 31 = 0, 32 25 80 64 52 101