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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 10.4.47f
Chapter 10, Problem 10.4.47f

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."

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Step 1: Understand the statement. It says: If the limit of the sequence \( a_k \) as \( k \to \infty \) is zero, then the infinite series \( \sum a_k \) converges. We need to determine if this is true or false.
Step 2: Recall the necessary condition for series convergence: For a series \( \sum a_k \) to converge, the terms \( a_k \) must approach zero as \( k \to \infty \). This means \( \lim_{k \to \infty} a_k = 0 \) is necessary but not sufficient for convergence.
Step 3: To test if the statement is true, consider a counterexample where \( \lim_{k \to \infty} a_k = 0 \) but \( \sum a_k \) diverges. A classic example is the harmonic series \( a_k = \frac{1}{k} \), which tends to zero but the series \( \sum \frac{1}{k} \) diverges.
Step 4: Explain why the harmonic series diverges despite its terms tending to zero. This shows that having terms approach zero does not guarantee the sum converges.
Step 5: Conclude that the statement is false because the limit of the terms being zero is necessary but not sufficient for the convergence of the series.

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Key Concepts

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

Limit of a Sequence

The limit of a sequence {aₖ} as k approaches infinity is the value that the terms of the sequence get arbitrarily close to. If lim (k → ∞) aₖ = 0, it means the terms become very small, but this alone does not guarantee the sum of the terms converges.
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Introduction to Sequences

Convergence of an Infinite Series

An infinite series ∑ aₖ converges if the sequence of its partial sums approaches a finite limit. Even if the terms aₖ approach zero, the series may diverge if the partial sums grow without bound.
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Convergence of an Infinite Series

Counterexample: Harmonic Series

The harmonic series ∑ 1/k is a classic example where the terms approach zero, but the series diverges. This shows that having terms tend to zero is necessary but not sufficient for series convergence.
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P-Series and Harmonic Series
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

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.

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.