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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.8.5
Chapter 10, Problem 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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1
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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Key Concepts

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

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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Related Practice
Textbook Question

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

Textbook Question

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

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

Textbook Question

Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

Textbook Question

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

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{tan⁻¹(10n⁄(10n + 4))}  

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Textbook Question

Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k. 

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