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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.83f
Chapter 10, Problem 10.2.83f

Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
f.If the sequence {aₙ} diverges, then the sequence {0.000001aₙ} diverges.

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1
Recall the definition of divergence for a sequence: a sequence \( \{a_n\} \) diverges if it does not approach a finite limit as \( n \to \infty \).
Consider the sequence \( \{0.000001 \cdot a_n\} \). This is the original sequence \( \{a_n\} \) multiplied by a constant scalar \( 0.000001 \).
Multiplying a sequence by a nonzero constant scales its terms but does not change whether the sequence converges or diverges. Specifically, if \( \{a_n\} \) diverges to infinity or oscillates without limit, then \( \{0.000001 \cdot a_n\} \) will also diverge (though possibly to a different infinite value or oscillation).
However, if \( \{a_n\} \) diverges because it oscillates or does not settle to a limit, scaling by \( 0.000001 \) will not make it converge; it will still fail to approach a finite limit.
Therefore, the statement is true: if \( \{a_n\} \) diverges, then \( \{0.000001 \cdot a_n\} \) also diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequence Divergence

A sequence diverges if it does not approach a finite limit as n approaches infinity. Divergence means the terms either grow without bound, oscillate, or fail to settle at any single value.
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Scalar Multiplication of Sequences

Multiplying each term of a sequence by a constant scales the sequence but does not necessarily preserve its convergence or divergence properties. The behavior depends on the original sequence and the scalar.
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Counterexamples in Mathematical Reasoning

A counterexample disproves a universal statement by providing a specific case where the statement fails. Using counterexamples is essential to test the validity of claims about sequences.
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Related Practice
Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval (1/2, ∞).

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."