Final answer:
The interval of convergence for the given function can be found by analyzing the power series representation. The series converges when the absolute value of x is less than 2. Therefore, the interval of convergence is (-2, 2).
Step-by-step explanation:
The interval of convergence can be found by analyzing the power series representation of the given function. The function is given as f(x) = x²/x⁴+16, which can be written as f(x) = x²/(x⁴+16). The power series representation can then be obtained by expanding the denominator using the geometric series formula. Since the series is convergent when the absolute value of x is less than the radius of convergence, we need to find the radius of convergence. The interval of convergence is then (-R, R), where R is the radius of convergence.
The power series expansion of the denominator can be written as 1/(1 - (-x⁴/16)), which is equivalent to the sum of the geometric series 1 + (-x⁴/16) + (-x⁴/16)² + ...
This geometric series converges when the absolute value of the ratio, |-x⁴/16|, is less than 1. Solving this inequality gives |-x⁴/16| < 1, which simplifies to |x⁴| < 16. Taking the fourth root of both sides, we get |x| < 2. This means that the series converges when the absolute value of x is less than 2.
Therefore, the interval of convergence is (-2, 2).