Question

Find a power series representation for the function f(x) = 8/(3 - x). Give your power series representation centered at x = 0.

Answer 1

To find the power series representation for f(x) = 8/(3 - x), we use the geometric series template and express f(x) as a series by substituting r with x/3 and multiplying by 8/3. The resulting power series is Σ (8/3^n+1 * x^n), converging for |x| < 3.

Step-by-step explanation:

To find a power series representation for the function f(x) = 8/(3 - x) centered at x = 0, we start by rewriting the function in a form that allows us to use a known power series. The geometric series 1/(1 - r) converges to a power series provided |r| < 1, where the series is Σ (r^n) from n = 0 to infinity. We express f(x) in a similar form:

f(x) = 8/(3 - x) = 8 * 1/(3(1 - x/3))

We see that the series can be represented by substituting r with x/3 and multiplying the known geometric series by 8/3:

• f(x) = 8/3 * Σ ((x/3)^n)

The power series centered at x = 0 is then:

• f(x) = Σ (8/3^n+1 * x^n)

This series converges for |x/3| < 1, which implies |x| < 3.