Find the radius of convergence and interval of convergence of the series
Well, that was the application of the Ratio Test.
You did not make any conclusion.
Why do you require simplification? Let's draw some conclusions without it. It looks to me like just about every piece is approaching unity. Where does that leave us?
Note: Nice screen name. I wish more folks felt that way about mathematics! Why do you hoist a China flag and speak from Viet Nam?
Other Note: From your experience, does whatever culture you know tell women and girls that they can't do math? Obviously, that would be a silly thing to tell you, but it may remain a common cultural message.