Question

With power series, is an endpoint convergent if you plug it back into the original series, and get an alternating series that is conditionally convergent?

Answer 1

The ratio test or the n-th root test does not indicate absolute convergence at the end-points.

In general, the radius of convergence is defined for absolute convergence, so if a test for the end-point(s) yield a conditional convergence, the radius of convergence remains an open interval, i.e. the end-points are excluded.