Math problem, power series Calculus2 for professionals

Question

Find the radius of convergence and interval of convergence of the following power series

∑ from n=1 to ∞ [ (X-1)^n] ÷[(3^n) *(n+1)^3]

PLEASE SOLVE WITH STEPS

To the final result because whenever I solve, I get different answer.

Answer 1

Use the ratio test which will result in a ratio of |(1/3)(x-1)[(n+1)/(n+2)]³|.

The term in square brackets goes to 1 so that in order for convergence

|(x-1)|<3. I will leave it to you to answer the specific questions.