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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.4.39c
Capitolo 10, Problema 10.4.39c

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

Guida verificata passo dopo passo
1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^7} \). This is a p-series with \( p = 7 > 1 \), so it converges.
Recall Theorem 10.13, which states that for a convergent series with positive, decreasing terms, the remainder \( R_n = S - S_n \) (the error when approximating the sum by the first \( n \) terms) 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. Here, \( S_n = \sum_{k=1}^n \frac{1}{k^7} \) is the partial sum up to \( n = 2 \).
Calculate the integrals to find the bounds for the remainder:
\[ \int_n^{\infty} \frac{1}{x^7} \, dx \quad \text{and} \quad \int_{n+1}^{\infty} \frac{1}{x^7} \, dx. \]
Finally, use these bounds to write the inequalities for the exact sum \( S \):
\[ S_n + \int_{n+1}^{\infty} \frac{1}{x^7} \, dx \leq S \leq S_n + \int_n^{\infty} \frac{1}{x^7} \, dx. \]

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Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. Understanding convergence ensures that the series has a well-defined sum, which is essential when estimating bounds for the series' exact value.
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Convergence of an Infinite Series

Theorem 10.13 (Bounds for Series Sums)

Theorem 10.13 provides a method to find lower and upper bounds for the sum of a convergent series using partial sums and remainder estimates. It typically involves comparing the remainder to an integral or another expression to bound the error after n terms.
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Intro to Series: Partial Sums

Partial Sums and Remainder Estimation

Partial sums sum the first n terms of a series, approximating the total sum. The remainder is the difference between the exact sum and the partial sum. Estimating this remainder allows us to find bounds (Lₙ and Uₙ) that enclose the true sum.
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Estimating the Area Under a Curve with Right Endpoints & Midpoint
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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.


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.

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{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.



c.Find the limit of the sequence. What is the physical meaning of this limit?

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Explain why or why not

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


c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

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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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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.


{-5, 5, -5, 5, ......}

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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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