Find the radius of convergence and interval of convergence of the following power series ∑n=1 ∞ (X-1)^n ÷ 3n(n+1)^3

Question

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

series

∑_n=1 ^∞ (X-1)^n ÷ 3n(n+1)^3

PLEASE SOLVE WITH STEPS

Answer 1

Taha:

Use a_{n =} (x-1)ⁿ/(3*n*(n+1)³)

a_n+1 = (x-1)ⁿ⁺¹/(3*(n+1)*(n+2)³)

|a_n+1/a_n| = | (x-1) * (n/n+1) * ((n+1)/(n+2))³ |

as n goes to ∞ |a_n+1/a_n| goes to |x-1|

We need to find out on what interval |x-1|<1

Solve this inequality and don't forget to test the endpoints. Let me know if you need further help!