Are there any results in Rudin that we are not allowed to be used at this point? E.g. L'Hospitals rule and derivative of the logarithm?
Yes! Those not in the chapter of series :). But if you do not find another way go ahead.
But remember that you can always use result you have seen in previous courses :).
I am assuming you wanted to use result ahead to compute the limits giving the radii of convergence....
I am assuming you wanted to use result ahead to compute the limits giving the radii of convergence....
Now it should be ok!