Question

Is it possible for an alternating series to converge absolutely but not conditionally?

Accepted Answer

Recall the definitions: An alternating series is a series whose terms alternate in sign, typically written as $$\sum (-1)^n a_n$$ with $$a_n \u003e 0. $$Absolute convergence means $$\sum |a_n|$$ converges, while conditional convergence means $$\sum a_n$$ converges but $$\sum |a_n|$$ diverges. Understand the relationship between absolute and conditional convergence: If a series converges absolutely, it means the series of absolute values converges. This implies the original series also converges (since absolute convergence is a stronger condition). Analyze the question: Can an alternating series converge absolutely but not conditionally? Since absolute convergence implies convergence of the original series, the original series must also converge. Therefore, if it converges absolutely, it automatically converges (not just conditionally, but unconditionally). Conclude that an alternating series cannot converge absolutely but fail to converge conditionally, because absolute convergence guarantees convergence of the original series, making conditional convergence irrelevant in that case. Summarize: Absolute convergence implies convergence of the original series, so an alternating series that converges absolutely also converges (not just conditionally). Hence, it is not possible for an alternating series to converge absolutely but not conditionally.

Is it possible for an alternating series to converge absolutely but not conditionally?

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Concetti chiave

Absolute Convergence

Conditional Convergence

Relationship Between Absolute and Conditional Convergence