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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 10.4.41d
Chapter 10, Problem 10.4.41d

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


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


41. ∑ (k = 1 to ∞) 1 / k⁶

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1
Recognize that the series given is \( \sum_{k=1}^{\infty} \frac{1}{k^6} \), which is a convergent p-series with \( p = 6 > 1 \). This means the series converges absolutely and the remainder after \( n \) terms can be estimated using the integral test remainder bounds.
To approximate the sum using the first 10 terms, calculate the partial sum \( S_{10} = \sum_{k=1}^{10} \frac{1}{k^6} \). This is the sum of the first 10 terms of the series.
Use the integral test remainder estimate to find bounds for the remainder \( R_{10} = S - S_{10} \), where \( S \) is the total sum of the series. The remainder satisfies the inequalities: \[ \int_{11}^{\infty} \frac{1}{x^6} \, dx \leq R_{10} \leq \int_{10}^{\infty} \frac{1}{x^6} \, dx \] These integrals provide lower and upper bounds for the error when approximating the infinite sum by the first 10 terms.
Evaluate the improper integrals: \[ \int_{a}^{\infty} \frac{1}{x^6} \, dx = \left[ -\frac{1}{5x^5} \right]_{a}^{\infty} = \frac{1}{5a^5} \] Use this formula to compute the bounds for \( R_{10} \) by substituting \( a = 10 \) and \( a = 11 \).
Finally, add these remainder bounds to the partial sum \( S_{10} \) to get the interval in which the total sum \( S \) lies: \[ S_{10} + \int_{11}^{\infty} \frac{1}{x^6} \, dx \leq S \leq S_{10} + \int_{10}^{\infty} \frac{1}{x^6} \, dx \] This interval gives a guaranteed range for the value of the series when approximated by the first 10 terms.

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Key Concepts

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

Convergent Infinite Series

A convergent infinite series is a sum of infinitely many terms that approaches a finite limit. For the series ∑ 1/k⁶, since the exponent 6 > 1, the p-series test confirms convergence. Understanding convergence ensures the series sum exists and can be approximated by partial sums.
Recommended video:
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Convergence of an Infinite Series

Partial Sums and Approximation

Partial sums are the sums of the first n terms of a series and serve as approximations to the total infinite sum. Using ten terms means calculating the sum from k=1 to 10, which approximates the series' value. The accuracy depends on how quickly the remaining terms decrease.
Recommended video:
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Introduction to Riemann Sums

Remainder (Error) Estimates for Series

The remainder after n terms is the difference between the infinite sum and the nth partial sum. For decreasing positive term series like ∑ 1/k⁶, the remainder can be bounded using the integral test, providing an interval where the true sum lies. This helps estimate the error in approximations.
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Alternating Series Remainder
Related Practice
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

Textbook Question

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

d. When applying the Limit Comparison Test, an appropriate comparison series for ∑ (k = 1 to ∞) (k² + 2k + 1) / (k⁵ + 5k + 7) is ∑ (k = 1 to ∞) 1 / k³.

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

Textbook Question

87. Explain why or why not

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


d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

Textbook Question

Explain why or why not

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


d.If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and

{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},

then limₙ→∞aₙ = limₙ→∞bₙ.

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Textbook Question

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


d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².