Final Answer:
The power series representation for the function f(x) = x⁶/(9x⁵ - 2) centered at x = 0 is Σₙ (x^(5n+6))/(2⋅9ⁿ).
Step-by-step explanation:
To determine the power series representation of f(x) centered at x = 0, the geometric series formula is applied. Initially, express the function in a form where x is raised to a power. For f(x) = x⁶/(9x⁵ - 2), it can be rewritten as f(x) = x^(5n+6)/(2⋅9ⁿ) by factoring x⁵ from the denominator.
Utilizing the geometric series formula, which states that 1/(1-r) = Σₙ rⁿ when |r| < 1, the variable r is identified as -1/(18)x⁵. As |(-1/(18))x⁵| < 1, the formula is applied, yielding the power series representation:
f(x) = x⁶/(9x⁵ - 2) = x^(5n+6)/(2⋅9ⁿ) = Σₙ (x^(5n+6))/(2⋅9ⁿ).
This power series is valid within the convergence radius of |x| < (2/9)^(1/5)⋅3, ensuring its applicability in a certain range. The power series form offers a convenient means to represent the given function as an infinite sum, facilitating mathematical analyses and approximations in various contexts.