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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 10.4.41b
Chapter 10, Problem 10.4.41b

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.


41. ∑ (k = 1 to ∞) 1 / k⁶

Verified step by step guidance
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Recognize that the series given is \( \sum_{k=1}^{\infty} \frac{1}{k^6} \), which is a convergent p-series with \( p = 6 > 1 \).
To estimate the remainder \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^6} \), use the integral test remainder estimate, which states that \( R_n \leq \int_n^{\infty} \frac{1}{x^6} \, dx \).
Set up the integral \( \int_n^{\infty} x^{-6} \, dx \) and evaluate it: \( \int_n^{\infty} x^{-6} \, dx = \lim_{t \to \infty} \int_n^t x^{-6} \, dx \).
Calculate the definite integral: \( \int_n^t x^{-6} \, dx = \left[ \frac{x^{-5}}{-5} \right]_n^t = \frac{1}{5 n^5} - \lim_{t \to \infty} \frac{1}{5 t^5} \). Since \( \lim_{t \to \infty} \frac{1}{t^5} = 0 \), the integral equals \( \frac{1}{5 n^5} \).
Set the remainder estimate less than \( 10^{-3} \): \( \frac{1}{5 n^5} < 10^{-3} \). Solve this inequality for \( n \) to find the minimum number of terms needed.

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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 the series ∑ 1/k⁶, since the exponent 6 > 1, it converges by the p-series test. Understanding convergence ensures the remainder (error) after a finite number of terms is well-defined and can be estimated.
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Convergence of an Infinite Series

Remainder (Error) in Infinite Series

The remainder after n terms is the difference between the infinite sum and the nth partial sum. For positive, decreasing terms, the remainder can be bounded to estimate how many terms are needed to achieve a desired accuracy, such as ensuring the remainder is less than 10⁻³.
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Convergence of an Infinite Series

Integral Test and Remainder Estimate

The integral test compares a series to an improper integral to determine convergence and estimate remainders. For decreasing positive functions, the remainder after n terms is less than the integral from n to infinity of the function, providing a practical way to find the number of terms needed for a given error bound.
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Integral Test
Related Practice
Textbook Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.


Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.


b.Find a recurrence relation that generates the sequence {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.



b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.

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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

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

Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 30,r = 0.25

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


b.Find an explicit formula for the terms of the sequence.


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.

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