Final answer:
To expand the function f(x) = 9/(1+3x^4) in a power series with center c=0, we can use the geometric series formula. The interval of convergence is -1 < x < 1.
Step-by-step explanation:
To expand the function f(x) = 9/(1+3x^4) in a power series with center c=0, we can use the geometric series formula. The formula states that if |x| < 1, then 1/(1+x) can be expressed as a power series: 1/(1+x) = 1 - x + x^2 - x^3 + x^4 - ...
Using this formula, we can rewrite f(x) as a power series: f(x) = 9(1 - 3x^4 + 9x^8 - 27x^12 + ...)
The interval of convergence can be determined by analyzing the convergence of the power series. In this case, the power series converges when |x^4| < 1, which means -1 < x^4 < 1. Taking the fourth root of these inequalities, we get -1 < x < 1. Therefore, the interval of convergence is -1 < x < 1.