Determine the radius of convergence of the power series:
where and . Test for convergence at the boundary points if is finite.
First, we write
where the sum on the left converges if and only if both sums on the right converge (since and these are all positive terms; hence, we cannot have their sum convergent while either of the individual series are divergent since they would diverge to ). So, we apply the root test to the two series on the right:
Thus, the first series on the right converges for and the second converges for . Therefore, the series on the left converge for