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

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

1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯

Textbook Question

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.

aₙ = (−1)ⁿ / 2ⁿ

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.

37.1 + e/π + e²/π² + e³/π³ + ⋯

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Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

47.0.3̅ = 0.333…

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Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / lnk

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(1/k)