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.

Accepted Answer

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.

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.

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Key Concepts

Absolute Convergence

Conditional Convergence

Divergence and Counterexamples