I'm having trouble with the algebraic manipulation required to find the interval of convergence of many of the Power series and Taylor series problems in my calculus class.

For example:

Using the Ratio test, and ignoring because we can, we have:

The can be cancelled:

Distributing the :

Cancelling the terms:

The next step:

I want to say that the limit is infinity but I think that would mean the interval of convergence is all real numbers; I don't think that is the answer. Perhaps I made algebraic/arithmetic mistake or perhaps I am not considering the limit correctly.