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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 10.4.39b
Chapter 10, Problem 10.4.39b

39–40. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, use Theorem 10.13 to complete the following.


b. Find an upper bound for the remainder Rₙ.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Verified step by step guidance
1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^7} \). This is a p-series with \( p = 7 \), which converges because \( p > 1 \).
Recall Theorem 10.13 (the Integral Test remainder estimate), which states that for a decreasing, positive, continuous function \( f(k) \), the remainder \( R_n = S - S_n \) satisfies the inequality \( R_n \leq \int_{n}^{\infty} f(x) \, dx \).
Set \( f(x) = \frac{1}{x^7} \) and write the upper bound for the remainder as \( R_n \leq \int_{n}^{\infty} \frac{1}{x^7} \, dx \).
Evaluate the improper integral \( \int_{n}^{\infty} x^{-7} \, dx \) by finding the antiderivative of \( x^{-7} \), which is \( \frac{x^{-6}}{-6} \), and then compute the limit as the upper bound approaches infinity.
Substitute \( n = 2 \) into the evaluated integral to express the upper bound for the remainder \( R_2 \) explicitly in terms of \( n \).

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Key Concepts

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

Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k⁷ converges because the terms decrease rapidly and satisfy the p-series test with p = 7 > 1. Understanding convergence ensures that the remainder or error after n terms is well-defined.
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Convergence of an Infinite Series

Remainder (Error) in a Series Approximation

The remainder Rₙ is the difference between the infinite series sum and the partial sum up to n terms. It measures the error when approximating the series by a finite number of terms. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.
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Alternating Series Remainder

Theorem 10.13 (Integral Test Remainder Estimate)

Theorem 10.13 provides bounds for the remainder of a convergent series whose terms come from a positive, decreasing function. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This theorem allows calculation of explicit upper bounds for the error.
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Integral Test
Related Practice
Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


{1, 3, 9, 27, 81, ......}

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Textbook Question

Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation

aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

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Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. A series that converges absolutely must converge.

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