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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 10.4.43b
Chapter 10, Problem 10.4.43b

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

Verified step by step guidance
1
Identify the series given: it is a geometric series with the general term \( a_k = \frac{1}{3^k} \).
Recall that for a geometric series \( \sum_{k=1}^\infty ar^{k-1} \), the sum to infinity is \( \frac{a}{1-r} \) if \( |r| < 1 \). Here, the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \).
The remainder \( R_n \) after summing the first \( n \) terms is the sum of the terms from \( n+1 \) to infinity. For a geometric series, \( R_n = \frac{a r^n}{1-r} \). In this problem, \( R_n = \frac{\frac{1}{3} \left( \frac{1}{3} \right)^n}{1 - \frac{1}{3}} \).
Simplify the expression for \( R_n \) to get a formula in terms of \( n \): \( R_n = \frac{\frac{1}{3^{n+1}}}{1 - \frac{1}{3}} = \frac{\frac{1}{3^{n+1}}}{\frac{2}{3}} = \frac{1}{2 \cdot 3^n} \).
Set the remainder \( R_n \) less than \( 10^{-3} \) and solve the inequality \( \frac{1}{2 \cdot 3^n} < 10^{-3} \) 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.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. For |r| < 1, the infinite series converges to a finite sum S = a / (1 - r), where a is the first term. Understanding this helps identify the sum and behavior of the series ∑ 1/3^k.
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Geometric Series

Remainder (Error) in Infinite Series

The remainder after n terms in a convergent series is the difference between the infinite sum and the partial sum up to n terms. For geometric series, the remainder can be explicitly calculated as R_n = S - S_n = a * r^n / (1 - r), which helps estimate how many terms are needed to achieve a desired accuracy.
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Convergence of an Infinite Series

Convergence Criteria and Error Bounds

To ensure the remainder is less than a specified tolerance (e.g., 10⁻³), one must use the error bound formula for the series. This involves solving inequalities involving the remainder expression to find the minimum number of terms n that satisfy the accuracy requirement.
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Determining Error and Relative Error
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

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

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)?

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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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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

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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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