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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 10.4.39a
Chapter 10, Problem 10.4.39a

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.


a. Use Sₙ to estimate the sum 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 \( p = 7 \), which converges because \( p > 1 \).
Recall Theorem 10.13, which provides bounds for the remainder \( R_n = S - S_n \) of a convergent series with positive, decreasing terms. The theorem states that the remainder 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. To estimate the sum \( S \), first compute the partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^7} \) for \( n = 2 \).
Next, calculate the integrals to find the bounds for the remainder:
\[ \int_3^{\infty} \frac{1}{x^7} \, dx \quad \text{and} \quad \int_2^{\infty} \frac{1}{x^7} \, dx. \]
Use these integral values to find the lower and upper bounds for the total sum \( S \) by adding them to \( S_2 \):
\[ S_2 + \int_3^{\infty} \frac{1}{x^7} \, dx \leq S \leq S_2 + \int_2^{\infty} \frac{1}{x^7} \, dx. \]

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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. For example, the series ∑ 1/k^7 converges because the terms decrease rapidly and satisfy the p-series test with p = 7 > 1. Understanding convergence ensures that the sum S exists and can be approximated.
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Convergence of an Infinite Series

Partial Sums (Sₙ)

The partial sum Sₙ is the sum of the first n terms of a series and serves as an approximation to the total sum. For the series ∑ 1/k^7, S₂ = 1 + 1/2^7 provides an estimate of the infinite sum. Partial sums are fundamental in estimating series values and analyzing error bounds.
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Integration Using Partial Fractions

Theorem 10.13 (Bounds on Remainder of a Series)

Theorem 10.13 gives lower and upper bounds for the remainder (error) when approximating a convergent series by its partial sum Sₙ. It typically uses integrals to bound the tail of the series, allowing us to estimate how close Sₙ is to the actual sum. This theorem is essential for quantifying the accuracy of partial sums.
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Alternating Series Remainder
Related Practice
Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.


∑ (k = 0 to ∞) (–2/7)ᵏ

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Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

a. Use a telescoping series argument.

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Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Textbook Question

Explain why or why not

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

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

Textbook Question

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 20,r = 0.5