Find the 4th Taylor polynomial, P4, generated by f(x) = 1/x at center a = 2 ?

Answer 1

just plug and chug

f(x) = f(2) + f'(2)(x-2)/1! + ...
f(2) = 1/2
f'(2) = -1/x^2 = -1/2^2 = -1/4
f"(2) = 2/x^3 = 1/4
f⁽³⁾(2) = -6/x^4 = -3/8
f⁽⁴⁾(2) = 24/x^4 = 3/4

f(x) = 1/2 + (-1/4)(x-2)/1! + (1/4)(x-2)^2/2! + (-3/8)(x-2)^3/3! + (3/4)(x-2)^4/4! + ...
= 1/2 - 1/4 (x-2) + 1/8 (x-2)^2 - 1/16 (x-2)^3 + 1/32 (x-2)^4 + ...