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?

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.

To determine if an endpoint is convergent when plugging it back into the original power series and getting an alternating series, we need to apply the Alternating Series Test.

The Alternating Series Test states that if an alternating series satisfies two conditions, it is conditionally convergent:

1. The absolute value of each term in the series decreases as the index increases.
2. The limit of the absolute value of the terms of the series as the index approaches infinity is zero.

Let's go through the steps to apply the Alternating Series Test and determine if the endpoint is convergent:

1. Start with the power series you have, which is an infinite series representation of a function.

2. Plug the endpoint value back into the series and obtain an alternating series. An alternating series has terms that alternate in sign, such as (-1)^n or (-1)^(n+1).

3. Check if the terms of the alternating series satisfy the two conditions of the Alternating Series Test. First, verify if the absolute value of each term is decreasing as the index increases. This condition ensures that the series is alternating. If the terms do not alternate in sign, then the Alternating Series Test does not apply, and we cannot determine convergence using this method.

4. Additionally, compute the limit of the absolute value of the terms as the index approaches infinity. If this limit is zero, then the series satisfies the second condition of the Alternating Series Test.

If both conditions of the Alternating Series Test are satisfied, the series is conditionally convergent. However, if either condition fails, the test is inconclusive, and we need to apply an alternate method to evaluate the convergence of the series.

Remember, plugging in an endpoint and obtaining an alternating series is just the first step. The convergence of the series still needs to be determined using the Alternating Series Test, as explained above.