I need to find radius of convergence and interval of convergence of

In d'Alembert feature I find radius like this

Radius is 1

If I try in Cauchy feature, I get same:

first square root

square root

Radius is 1

Interval of convergence:

I take -1 and 1

and

I do check if diverge/converge:

it diverge

here!=means'not even'

Interval is

Questions:

- Did I do task correct?
- it diverge or converge?
- How interval of convergence change because of divergence and convergence?