Final answer:
The power series representation for the function f(x) = x¹¹ / (x² - 1) is obtained by rewriting the function to make use of geometric series expansions, resulting in a series valid for |x| < 1.
Step-by-step explanation:
The function f(x) = x¹¹ / (x² - 1) can be expressed as a power series centered at x = 0 by first manipulating it into a form that can be expanded using a known series expansion, like the geometric series or the binomial theorem. We notice that x² - 1 is the difference of squares, which can be factored into (x - 1)(x + 1). Therefore, we consider f(x) as:
f(x) = x¹¹ / [(x - 1)(x + 1)].
To find the power series, the function must be in a form that allows it to be expanded around the center x = 0. Since the geometric series gives us 1/(1 - x) = ∞ Σ n=0 xⁿ, we rewrite f(x) to make use of this series expansion:
Let's split the fraction into two parts:
f(x) = x¹¹ / (x - 1)(x + 1) = x¹ / (1 - (-x)) - x¹ / (1 - x)
Now we can use the geometric series expansion:
f(x) = x¹¹( ∞ Σ n=0 (-x)ⁿ - ∞ Σ n=0 xⁿ)
The resulting power series for f(x) centered at x = 0 will be the sum of two geometric series. Remember that this power series representation is only valid for |x| < 1, as this is the radius of convergence for the geometric series.