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

In case of , bound points with

so, and converge.

And also,

\lvert \frac {a^n} {n} z^n + \frac {b^n} {n^2} z^n \rvert = \lvert \frac{1}{n} + \frac{(\frac{b}{a})^n}{n^2} \rvert = \frac{1}{n} + \frac{(\frac{b}{a})^n}{n^2} > \frac{1}{n}

\sum \lvert \frac {a^n} {n} z^n + \frac {b^n} {n^2} z^n \rvert

diverge

I think the boundary points converge as well- either all points by comparison with 1/n^2, or all z≠1 by dirichlet depending on b or a being larger, respectively.

The boundary points do converge. The following outline explains Tom’s comment a bit more:\\

For the case where , let and compare the magnitude of the series with . The series absolutely converges on the boundary.\\

For the case where , let and use Dirichlet’s test along with theorem 10.19, to show that the series converges for .\\