Question

Find a power series for f(x)=9x−3x²−1 centered at 0 by adding two power series

Answer 1

The power series for the function f(x) = 9x - 3x² - 1 centered at 0 is simply the function itself written out as terms involving varied powers of x: -1 + 9x - 3x². Since it is a polynomial, it is already in power series form.

Step-by-step explanation:

To find a power series for the function f(x) = 9x - 3x² - 1 centered at 0, we can write each term as a separate power series and then add them together. The constant term -1 is already in power series form, as it is equal to -1·x°. The linear term 9x can be written as 9·x¹, and the quadratic term -3x² as -3·x². Combining these, we have the power series:

f(x) = -1 + 9x - 3x²

This power series is already simplified and represents the function f(x) as an infinite sum with terms of increasing powers of x. No additional terms are needed because the function we are dealing with is a polynomial, which is already in the form of a finite power series.