Can you answer this calculus question?

Question

The series ∑ n!(x-3)^n from n=0 to infinity converges if and only if 

 

a) x=0

b) 2<x<4

c) x=3

d) 2≤x≤4

e) x<2 or x>4

 

Answer: C

 

Please show all your work.

Answer 1

Hi sun!

 

Using the ratio test lim (n→∞)|(n+1)!(x-3)^(n+1))/(n!(x-3)^n )|=lim (n→∞)|(n+1)(x-3)|, the series converge iff lim (n→∞)|(n+1)(x-3)|<1, so x=3.