Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.4.39b

Capitolo 10, Problema 10.4.39b

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

Guida verificata passo dopo passo

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 (the Integral Test remainder estimate), which states that for a decreasing, positive, continuous function \( f(k) \), the remainder \( R_n = S - S_n \) satisfies the inequality \( R_n \leq \int_{n}^{\infty} f(x) \, dx \).

Set \( f(x) = \frac{1}{x^7} \) and write the upper bound for the remainder as \( R_n \leq \int_{n}^{\infty} \frac{1}{x^7} \, dx \).

Evaluate the improper integral \( \int_{n}^{\infty} x^{-7} \, dx \) by finding the antiderivative of \( x^{-7} \), which is \( \frac{x^{-6}}{-6} \), and then compute the limit as the upper bound approaches infinity.

Substitute \( n = 2 \) into the evaluated integral to express the upper bound for the remainder \( R_2 \) explicitly in terms of \( n \).

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Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k⁷ converges because the terms decrease rapidly and satisfy the p-series test with p = 7 > 1. Understanding convergence ensures that the remainder or error after n terms is well-defined.

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06:52

Convergence of an Infinite Series

Remainder (Error) in a Series Approximation

The remainder Rₙ is the difference between the infinite series sum and the partial sum up to n terms. It measures the error when approximating the series by a finite number of terms. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.

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06:32

Alternating Series Remainder

Theorem 10.13 (Integral Test Remainder Estimate)

Theorem 10.13 provides bounds for the remainder of a convergent series whose terms come from a positive, decreasing function. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This theorem allows calculation of explicit upper bounds for the error.

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07:25

Integral Test

Pratica correlata

Domanda del libro di testo

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

b. (2n)! / (2n − 1)! = 2n

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Domanda del libro di testo

Explain why or why not

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

b.If a sequence of positive numbers converges, then the sequence is decreasing.

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Domanda del libro di testo

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 = 30,r = 0.25

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Domanda del libro di testo

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)]

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Domanda del libro di testo

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

h₀ = 20,r = 0.5

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