Determine whether the Maclaurin series converges absolutely, converges conditionally, or diverges at x = 1.

Question

This is a question from AP Calculus BC practice test.

I know this series is the Maclaurin Series for ${ln(x+1)}$ and it would converge on (-1, 1]. I am confused by the phrasing of question b).

It says determine Determine whether the Maclaurin series converges absolutely, converges conditionally, or diverges at x = 1. I understand that an entire series might converge conditionally, but how can a series converge conditionally at x = 1, a fixed value, when it is simply a converge or diverge situation?

Sorry that my question is more about the use of words rather than the question itself.

Answer 1

Converges absolutely means that the sum of the absolute values of the individual terms converges, but converges conditionally means that it converges, but not absolutely.

If you plug in the series at $x=1$ and use the p-series test, you can see whether it converges absolutely or conditionally.

Answer 2

Since you do have a series $\sum a_n$, you can always ask whether the series of absolute values $\sum \mid a_n\mid$ is convergent.

If so, it is absolutely convergent (and then the series converges as well). If not and the original series converges, it is conditionally convergent. (For instance, $\sum_{n=1}^\infty (-1)^n\dfrac1n$.)

Otherwise the series is divergent.