Final answer:
To find the radius of convergence, we can use the ratio test. The series converges for all values of x, meaning the radius of convergence is infinity.
Step-by-step explanation:
To find the radius of convergence, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. If the limit is greater than 1, the series diverges.
In this case, we have the series n*4^n*(x+2)^n. To apply the ratio test, we take the absolute value and divide the n+1 terms by the nth terms and simplify: |(n+1)*4^(n+1)*(x+2)^(n+1)| / (n*4^n*(x+2)^n) = |4(x+2)/(n+1)| Taking the limit as n approaches infinity.
We get: lim (n->inf) |4(x+2)/(n+1)| = 4|x+2|lim (n->inf) 1/(n+1) = 0. Since the limit is less than 1 for all values of x, the series converges for all values of x. Therefore, the radius of convergence, r, is infinity.