Ok let be a sequence in . Ok let us split this up into three cases: , , and .

Case 1

Suppose we have a series of the form . Then to determine the radius of convergence I apply the Ratio test. So . So now by our assumption above this is equivalent to , thus the radius of convergence of case 1 is and we are done.

Case 2

Now suppose we have the same series but we know have that , which is an identity for . Thus the radius of convergence is .

Case 3

Now once again we reach that which only occurs when .

The last two I did more holistically than the first, if you need more rigor I leave that up to you.