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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.6.34
Chapter 10, Problem 10.6.34

Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]

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1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2 + 2n + 1} \). Notice that the denominator can be factored as \( (n+1)^2 \).
Check for absolute convergence by considering the series of absolute values: \( \sum_{n=1}^{\infty} \left| \frac{(-1)^{n-1}}{(n+1)^2} \right| = \sum_{n=1}^{\infty} \frac{1}{(n+1)^2} \). This is a p-series with \( p = 2 \), which is known to converge.
Since the series of absolute values converges, the original series converges absolutely. Absolute convergence implies convergence.
For completeness, recognize that the original series is an alternating series because of the factor \( (-1)^{n-1} \). The terms \( \frac{1}{(n+1)^2} \) decrease to zero as \( n \to \infty \), satisfying the conditions of the Alternating Series Test.
Summarize: The series converges absolutely (and hence converges) because the absolute value series is a convergent p-series with \( p > 1 \).

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Key Concepts

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

Absolute Convergence

A series ∑aₙ converges absolutely if the series of absolute values ∑|aₙ| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it often simplifies analysis by allowing the use of comparison tests on positive terms.
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Conditional Convergence

A series ∑aₙ converges conditionally if it converges, but does not converge absolutely. This means the series converges due to the alternating signs or specific term behavior, but the series of absolute values diverges, highlighting the importance of sign alternation.
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Alternating Series Test

The Alternating Series Test states that an alternating series ∑(-1)ⁿbₙ converges if the sequence bₙ is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.
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Related Practice
Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) cos²n / n^(3/2)

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Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 8^(1/n)

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Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (2n + 2)! / (2n − 1)!

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Textbook Question

Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

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Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 1 to ∞) [3ⁿ / n³]

Textbook Question

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1