Final answer:
To find the center and radius of convergence of the power series, we need to find the values of x for which the series converges. The center is x = 3 and the radius of convergence is 1. The series converges if |x - 3| < 1.
Step-by-step explanation:
The power series ∑(n / 4^n(n + 1))(x - 3)^n can be written as a geometric series. A geometric series has the form ∑(a*r^n), where a is the first term and r is the common ratio. In this case, a = n / 4^n(n + 1) and r = (x - 3). To find the center and radius of convergence of the power series, we need to find the values of x for which the series converges.
The series converges if the absolute value of the common ratio (|r|) is less than 1. So, |x - 3| < 1. To find the center, we set x - 3 = 0 and solve for x. The center is x = 3. The radius of convergence is the distance from the center to the nearest point where the series diverges. In this case, the series diverges when x - 3 = 1 or x - 3 = -1. So, the radius of convergence is 1.