Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.5.5

Chapter 10, Problem 10.5.5

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

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Identify the general term of the series: $a_k = \frac{2^k}{3^k + 1}$.

To apply the Comparison Test, find a simpler series $b_k$ that resembles $a_k$ for large $k$ and whose convergence behavior is known.

Since \$3^k$ dominates the $+1$ in the denominator for large $k$, approximate $a_k$ by $\frac{2^k}{3^k}$.

Simplify the approximation: $\frac{2^k}{3^k} = \left(\frac{2}{3}\right)^k$.

Use the geometric series $\sum_{k=1}^\infty \left(\frac{2}{3}\right)^k$ as the comparison series, because it converges (common ratio $\frac{2}{3} < 1$).

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Key Concepts

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

Comparison Test

The Comparison Test determines the convergence or divergence of a series by comparing it to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than those of a divergent series, it diverges.

Geometric Series

A geometric series has terms of the form ar^k, where r is the common ratio. It converges if |r| < 1 and diverges otherwise. Recognizing geometric behavior helps in choosing a suitable comparison series for convergence tests.

Asymptotic Behavior of Terms

Analyzing the dominant terms in the numerator and denominator for large k helps simplify the series term. For example, 2^k/(3^k + 1) behaves like (2/3)^k for large k, guiding the choice of a comparison series with similar asymptotic behavior.

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Related Practice

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

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Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) cos(k) / k³

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Textbook Question

Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.

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Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

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Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) k / ln k

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Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²

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