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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 10.4.39c
Chapter 10, Problem 10.4.39c

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.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.


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 \( p = 7 > 1 \), so it converges.
Recall Theorem 10.13, which states that for a convergent series with positive, decreasing terms, the remainder \( R_n = S - S_n \) (the error when approximating the sum by the first \( n \) terms) is bounded by the integral test inequalities:
\[ \int_{n+1}^{\infty} f(x) \, dx \leq R_n \leq \int_n^{\infty} f(x) \, dx, \]
where \( f(x) = \frac{1}{x^7} \) in this problem. Here, \( S_n = \sum_{k=1}^n \frac{1}{k^7} \) is the partial sum up to \( n = 2 \).
Calculate the integrals to find the bounds for the remainder:
\[ \int_n^{\infty} \frac{1}{x^7} \, dx \quad \text{and} \quad \int_{n+1}^{\infty} \frac{1}{x^7} \, dx. \]
Finally, use these bounds to write the inequalities for the exact sum \( S \):
\[ S_n + \int_{n+1}^{\infty} \frac{1}{x^7} \, dx \leq S \leq S_n + \int_n^{\infty} \frac{1}{x^7} \, dx. \]

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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. Understanding convergence ensures that the series has a well-defined sum, which is essential when estimating bounds for the series' exact value.
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Convergence of an Infinite Series

Theorem 10.13 (Bounds for Series Sums)

Theorem 10.13 provides a method to find lower and upper bounds for the sum of a convergent series using partial sums and remainder estimates. It typically involves comparing the remainder to an integral or another expression to bound the error after n terms.
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Intro to Series: Partial Sums

Partial Sums and Remainder Estimation

Partial sums sum the first n terms of a series, approximating the total sum. The remainder is the difference between the exact sum and the partial sum. Estimating this remainder allows us to find bounds (Lₙ and Uₙ) that enclose the true sum.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
Textbook Question

Explain why or why not

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


c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.

It follows that limₙ→∞aₙ = limₙ→∞bₙ.

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Textbook Question

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.



c.Find the limit of the sequence. What is the physical meaning of this limit?

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Textbook Question

Explain why or why not

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


c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

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Textbook Question

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

c. Find an explicit formula for the nth term of the sequence.


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

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Textbook Question

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

c. Find an explicit formula for the nth term of the sequence.


{-5, 5, -5, 5, ......}

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Textbook Question

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


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


41. ∑ (k = 1 to ∞) 1 / k⁶

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