Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.R.11a

Chapter 10, Problem 10.R.11a

a.Does the sequence { k/(k + 1) } converge? Why or why not?

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Identify the general term of the sequence, which is given by \(a_k = \frac{k}{k + 1}\).

Recall that to determine if a sequence converges, we need to find the limit of \(a_k\) as \(k\) approaches infinity, i.e., compute \(\lim_{k \to \infty} \frac{k}{k + 1}\).

To find this limit, divide the numerator and denominator by \(k\) to simplify the expression: \(\frac{k}{k + 1} = \frac{\frac{k}{k}}{\frac{k + 1}{k}} = \frac{1}{1 + \frac{1}{k}}\).

Evaluate the limit of the simplified expression as \(k\) approaches infinity: since \(\frac{1}{k} \to 0\), the expression approaches \(\frac{1}{1 + 0} = 1\).

Conclude that since the limit exists and equals 1, the sequence \(\left\{ \frac{k}{k + 1} \right\}\) converges, and its limit is 1.

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Key Concepts

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

Definition of a Sequence

A sequence is an ordered list of numbers defined by a specific formula for its terms. Understanding how each term is generated, such as k/(k + 1), is essential to analyze its behavior as k increases.

Limit of a Sequence

The limit of a sequence is the value that the terms approach as the index k goes to infinity. If the terms get arbitrarily close to a fixed number, the sequence converges to that limit.

Convergence Criteria

A sequence converges if its limit exists and is finite. To determine convergence, evaluate the limit of the general term as k approaches infinity and check if it approaches a specific number.

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Related Practice

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)

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Textbook Question

Explain why or why not

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

a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

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Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

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Textbook Question

b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

Textbook Question

Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.