Final answer:
To express the function given as a power series, partial fractions are used to decompose the function, followed by expressing each term as a power series and finding the interval of convergence.
Step-by-step explanation:
Expressing a Function as a Power Series Using Partial Fractions
To express the function f(x) = 14 / (x² − 4x − 45) as the sum of a power series, we need to start by using partial fractions to decompose the rational expression into simpler fractions that can be easily written as power series. The first step is to factor the denominator x² − 4x − 45 which can be factored as (x − 9)(x + 5). Applying partial fractions, we can express f(x) as a sum of two fractions whose denominators are (x − 9) and (x + 5).
After determining the coefficients for the partial fractions, we can then use the geometric series formula to write each term as a power series. The next step is to find the radius of convergence for each series using the ratio test. Finally, the interval of convergence can be found which is the interval where the series converges to f(x).
It's important to note that each step of this process requires careful algebraic manipulation and a solid understanding of calculus and power series. The interval of convergence is particularly important because it tells us where the power series is a valid representation of the original function.