Necessay and Sufficient (slip the sums),
You can get rid of the absolute value because it takes on positives and negatives,
In the first summand, let (if then it converges, nothing is lost).
Thus, we can write,
The non-alternating term is,
Use the generalized ratio test,
Hence the reciprocal of this, which is 2, is the radius of absolute convergence.
Thus, (in both series)
Take reciprocal of both sides on the second one,
This is an annulus of absolute convergence.
I still think you need to check the boundary for convergence.
This is my 38th Post!!!