Find the radius of convergence and interval of convergence o

Find the radius of convergence and interval of convergence of the series sigma_n = 1^infinity (x - 2)^n/n^n

Solution

a_n=(x-2)ⁿ_/nⁿ

a_n+1=(x-2)ⁿ⁺¹_{/(n+1)ⁿ⁺¹}

ratio test for convergence:

lim_n->|a_n+1/a_n|<1

lim_n->|[(x-2)ⁿ⁺¹_{/(n+1)ⁿ⁺¹}]/[(x-2)ⁿ_/nⁿ]|<1

lim_n->|[(x-2)_{/(n+1)(1+(1/n))ⁿ}]|<1

|[0]|<1

radius of convergence =

interval of convergence is (-,)