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?

Answers

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.

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