Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
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14. Sequences & Series
Convergence Tests
5:44 minutesProblem 10.6.65d
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If ∑ aₖ diverges, then ∑ |aₖ| diverges.
Verified step by step guidance1
Recall the definitions: A series \( \sum a_k \) converges if the sequence of partial sums approaches a finite limit. It diverges if it does not. The series \( \sum |a_k| \) is called the series of absolute values, and if it converges, \( \sum a_k \) is said to converge absolutely.
The statement says: If \( \sum a_k \) diverges, then \( \sum |a_k| \) diverges. To analyze this, consider what absolute convergence means: If \( \sum |a_k| \) converges, then \( \sum a_k \) must also converge (absolutely convergent series always converge).
However, the converse is not necessarily true. A series can converge conditionally, meaning \( \sum a_k \) converges but \( \sum |a_k| \) diverges. This shows that \( \sum a_k \) converging does not imply \( \sum |a_k| \) converges.
The question is about divergence of \( \sum a_k \). If \( \sum a_k \) diverges, can \( \sum |a_k| \) converge? Consider a counterexample: The alternating harmonic series \( \sum (-1)^k \frac{1}{k} \) converges conditionally, but its absolute series \( \sum \frac{1}{k} \) diverges. This shows \( \sum a_k \) converges but \( \sum |a_k| \) diverges, which is not the case here, but it helps understand the relationship.
To directly address the statement, consider a series \( \sum a_k \) that diverges but \( \sum |a_k| \) converges. Is this possible? No, because if \( \sum |a_k| \) converges, then \( \sum a_k \) must converge absolutely, contradicting the divergence of \( \sum a_k \). Therefore, if \( \sum a_k \) diverges, \( \sum |a_k| \) must also diverge, making the statement true.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Convergence
A series ∑aₖ is absolutely convergent if the series of absolute values ∑|aₖ| converges. Absolute convergence implies convergence of the original series, but the converse is not necessarily true.
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Conditional Convergence
A series ∑aₖ is conditionally convergent if it converges, but the series of absolute values ∑|aₖ| diverges. This means the series converges only due to the specific arrangement of positive and negative terms.
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Divergence and Counterexamples
Divergence of ∑aₖ does not guarantee divergence of ∑|aₖ|. To determine the truth of the statement, one must consider counterexamples, such as series that diverge but whose absolute values converge or vice versa.
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