I've calculated the radius of convergence for in two different ways and I get two different answers.Find the range of values of values of x for which the series and converge, and find an expression for their sums, carefully justifying your answers.

The first way:

Let .

Therefore

Hence .

Alternatively, using the ratio test:

This tells me that .

Which one is correct.

For :

The main problem I have now is trying to find the expression for the sums. Can someone show me how to do that?