Determine the radius of convergence of the power series:

Test for convergence at the boundary points if is finite.

If then unless . Thus, the series does not converge for if .

If then we apply Dirichlet’s test where

is bounded for any (i.e., the partial sums are bounded) and the terms are monotonically decreasing with . Hence, the series converges for , which implies if .

If then for all so the series converges for all which implies .