Final answer:
To find a power series representation for the function f(x) = (5 + x) / (1 - x), we can use the geometric series sum formula. By expressing f(x) as a sum of two geometric series, we achieve the power series representation f(x) = 5 + 6x + 6x^2 + 6x^3 + ... for |x| < 1.
Step-by-step explanation:
We need to find a power series representation for the function f(x) = \frac{5+x}{1 - x}. This can be done using the geometric series sum formula, which is applicable because the function is similar to \frac{1}{1 - x}, the sum of a convergent geometric series with |x| < 1.
The geometric series sum formula is:
S = a + ar + ar^2 + ar^3 + ... = \frac{a}{1 - r}, for |r| < 1.
We can express f(x) in a similar form:
f(x) = 5 \times \frac{1}{1 - x} + \frac{x}{1 - x} = 5 \sum_{n=0}^{\infty} x^n + \sum_{n=0}^{\infty} x^{n+1}
Combining terms, we get the power series representation:
f(x) = \sum_{n=0}^{\infty} 5x^n + \sum_{n=1}^{\infty} x^n
Therefore, the power series for f(x) is:
f(x) = 5 + 6x + 6x^2 + 6x^3 + ... for |x| < 1.