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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 10.4.41c
Chapter 10, Problem 10.4.41c

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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Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{k^6} \) is a convergent p-series with \( p = 6 > 1 \), so it converges to a finite value.
To find lower and upper bounds \( L_n \) and \( U_n \) for the sum, consider the partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^6} \) as an approximation to the infinite sum.
Use the integral test remainder estimates: the remainder \( R_n = S - S_n \) satisfies \( \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \leq R_n \leq \int_n^{\infty} \frac{1}{x^6} \, dx \).
Evaluate the improper integrals \( \int_n^{\infty} \frac{1}{x^6} \, dx \) and \( \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \) to find explicit expressions for the upper and lower bounds on the remainder.
Add these remainder bounds to the partial sum \( S_n \) to get the inequalities: \( S_n + \int_{n+1}^{\infty} \frac{1}{x^6} \, dx \leq S \leq S_n + \int_n^{\infty} \frac{1}{x^6} \, dx \), which give the lower bound \( L_n \) and upper bound \( U_n \) for the exact sum.

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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.
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Convergence of an Infinite Series

Remainder (Error) Estimation in Series

Remainder estimation provides bounds on the difference between the partial sum and the exact sum of a convergent series. For positive, decreasing terms, the remainder after n terms is less than the next term or can be bounded using integral tests, helping to find upper and lower bounds.
Recommended video:
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Alternating Series Remainder

Integral Test for Series Bounds

The integral test compares a series to an improper integral to determine convergence and estimate remainders. For decreasing positive functions, the integral from n to infinity of f(x) dx bounds the remainder, allowing calculation of lower and upper bounds (Lₙ and Uₙ) on the series sum.
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Alternating Series Test
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.

Textbook Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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Textbook Question

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

b. Which of the following series converges faster? Explain.

∑ (k = 2 to ∞) 1 / (k(ln k)²) or ∑ (k = 3 to ∞) 1 / (k(ln k)(ln ln k)²)?

Textbook Question

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

Textbook Question

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


c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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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