Hello. I need math help. It says to find the radius of convergence and interval of convergence for:?

Question

(summation notation) n=0 to infinity of k!(x-1)^k. The radius is supposed to be = to 0, but I'm stuck on the limits part (convergence test). Can someone help?

Answer 1

Use the Ratio Test. Lim(k->inf)[(k+1)!(x-1)^(k+1)/k!(x-1)^k] = infinity. Therefore, the series converges only for x = 1, when all terms beyond the first are zero. There, the radius of convergence is 0, and the interval of convergence is [1,1].