Final answer:
The coefficients in the power series representation of the function f(x) = 9 - x are c_0 = 9, c_1 = c_2 = c_3 = c_4 = c_5 = 0. The radius of convergence R of the power series is infinity.
Step-by-step explanation:
To find the coefficients in the power series representation of the function f(x) = 9 - x, we can use the formula c_n = f^(n)(0)/n! where f^(n)(0) denotes the nth derivative of f(x) evaluated at x = 0. Since the function f(x) = 9 - x is a polynomial, all its derivatives are zero except for the constant term. Therefore, the first few coefficients are c_0 = 9, c_1 = 0, c_2 = 0, c_3 = 0, c_4 = 0, and c_5 = 0.
The radius of convergence R of the power series can be found using the formula R = 1/L, where L is the limit of the absolute value of the ratio of consecutive coefficients as n approaches infinity. In this case, since all the coefficients from c_1 to c_5 are zero, the limit of the ratio of consecutive coefficients is also zero. Therefore, the radius of convergence R is infinity.