Radius of convergence

[hgaon001]

I have two problems of radius of convergence

the sum from n=0 to infitinity of [(3^n)(x+2)^n]/(n+2)!

when i use the ratio test i get the lim n-> infinity (x+2)(n+2)/(n+3) i dont know that to do aftewards

also i have the sum from n=0 to infinity of [n!((x-3)^n)/4^(n+1)]

i get the limit as n->infinity [(n+1)(x-3)]/4

 

[fasteddie65]

[(3^n)(x+2)^n]/(n+2)! = a_n
|a_(n +1)/a_n| = |(3^(n+1)(x+2)^(n+1)/(n+3)! ÷ (3^n)(x+2)^n]/(n+2)!| = 3/(n+3) |x+2| --> 0 as n --> ?.

The radius of convergence is ?, i.e. it converges for all real numbers.

 

[fasteddie65]

[n!((x-3)^n)/4^(n+1)] = a_n

|a_(n+1)/a_n| = |(n+1)!(x - 3)^(n+1)/4^(n+2) ÷ n!(x-3)^n)/4^(n+1) = (n + 1)/4 |x - 3| --> ? as n --> ?, so the radius of convergence is 0, i.e. it convergence if and only if x - 3 = 0, or x = 3.