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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.5
Capitolo 10, Problema 10.8.5

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 10 to ∞) 1 / (k − 9)⁵

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First, rewrite the series to simplify the expression inside the summation. Notice that the term is \( \frac{1}{(k - 9)^5} \). Let \( n = k - 9 \), so when \( k = 10 \), \( n = 1 \). Thus, the series becomes \( \sum_{n=1}^{\infty} \frac{1}{n^5} \).
Recognize that the rewritten series is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = 5 \).
Recall the p-series convergence test: a p-series converges if and only if \( p > 1 \). Since \( p = 5 > 1 \), this series converges.
Therefore, the appropriate convergence test to identify this series' behavior is the p-series test.
No further simplification is necessary because the series is already in a standard form suitable for applying the p-series test.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges is fundamental. Convergence depends on the behavior of the terms as the index grows large.
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Convergence of an Infinite Series

p-Series Test

The p-series test applies to series of the form ∑ 1/n^p. It states that the series converges if p > 1 and diverges otherwise. Recognizing a series as a p-series or rewriting it into this form helps quickly determine convergence.
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P-Series and Harmonic Series

Index Shifting and Simplification

Rewriting or shifting the index of summation can simplify a series to a more recognizable form. For example, changing variables to start the index at 1 can help identify the series type and apply appropriate convergence tests.
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Writing a General Formula Example 1
Pratica correlata
Domanda del libro di testo

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

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

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84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.



{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3

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Domanda del libro di testo

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

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

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48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) (5 / 6)⁻ᵏ

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35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 1⁄10ⁿ; n = 1, 2, 3, …

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72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))

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