Your inequality is correct (though you may not necessarily have equality), but not the reason. The radius of convergence being one is the same as saying the expansion of converges whenever . In your case, which gives you
I suggest you take a look at this equation again. It should be . Then . When , we have (since a is positive, we drop the absolute value sign). This is true if and only if simply by manipulating the inequality.
Solving that I would get: