Tricky series radius of convergence question (analysis course)

[pcvt]

Homework Statement

Find the radius of convergence of sum from 1 to n of

1/(n^n) * x^(2^n)

Homework Equations

The Attempt at a Solution

Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent

 

[hunt_mat]

What about the nth root test?

 

[pcvt]

From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for

I believe its |x|<=1 but I'm not sure how to analytically prove this

 

[Dick]

From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for

I believe its |x|<=1 but I'm not sure how to analytically prove this

I don't think |x|=1 works. But the case |x|<1 is easy. And |x|>1 is not much harder, you just want to show it diverges somehow. I'd say the easiest way is to combine l'Hopital or a ratio test with a comparison test. For example, can you show 2^n/n>n for large enough values of n?