교과서 질문
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.
45
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
10장, 문제 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.
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
8:22
Introduction to Sequences
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.
추천 영상:
8:22Introduction to Sequences
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.
추천 영상:
가이드 코스
11:50Pumping Liquids Example 5
관련 실천
교과서 질문
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, ∞).
57
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교과서 질문
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.
38
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교과서 질문
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.
36
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교과서 질문
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."
63
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