Final answer:
The function f(x)=9x-3x^2-1 is already in the form of a power series as f(x) = -1 + 9x - 3x^2, with each term being a coefficient multiplied by the variable x raised to successive powers (0, 1, and 2) respectively.
Step-by-step explanation:
The given function f(x) = 9x - 3x^2 - 1 can be written as a power series by expressing it as a sum of two simpler power series, these being the geometric series and a linear term. Each term in the power series represents a coefficient multiplied by the variable x raised to a different power, starting from x to the power of 0 and increasing by one with each subsequent term.
Firstly, to express -3x^2 as a power series, we recognize that it is simply -3 times the power series of x^2, which is the second term of the series expansion for x^n.
By combining these, we get the final power series representation centered at 0: f(x) = -1 + 9x - 3x^2. This series is already in a power series form with terms x^0, x^1, and x^2 and does not require further expansion as it does not contain higher powers or other functions that need series representations.