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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 10.4.41a
Chapter 10, Problem 10.4.41a

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


a. Find an upper bound for the remainder in terms of n.


41. ∑ (k = 1 to ∞) 1 / k⁶

Verified step by step guidance
1
Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{k^6} \) is a convergent p-series with \( p = 6 > 1 \), which ensures convergence.
To find an upper bound for 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 improper integral \( \int_{n}^{\infty} x^{-6} \, dx \) to estimate the remainder.
Evaluate the integral: \( \int_{n}^{\infty} x^{-6} \, dx = \lim_{t \to \infty} \int_{n}^{t} x^{-6} \, dx \).
Compute the antiderivative of \( x^{-6} \), which is \( \frac{x^{-5}}{-5} \), then apply the limits from \( n \) to \( \infty \) to express the upper bound for the remainder \( R_n \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases. For series like ∑ 1/k⁶, which is a p-series with p > 1, convergence is guaranteed. Understanding convergence is essential to discuss remainders and error bounds.
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Convergence of an Infinite Series

Remainder (Error) in Partial Sums

The remainder after n terms of a convergent series is the difference between the infinite sum and the partial sum up to n. Estimating this remainder helps determine how close the partial sum is to the actual sum, which is crucial for approximations and error analysis.
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Integration Using Partial Fractions

Integral Test and Remainder Estimation

The integral test can be used to estimate the remainder of a decreasing positive term series by comparing the tail of the series to an improper integral. This provides an upper bound for the remainder, making it a practical tool for bounding errors in series approximations.
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Integral Test
Related Practice
Textbook Question

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯


89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.


a. Find the base-10 representation of the binary number 0.001̅1.

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Textbook Question

Explain why or why not

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

a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

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

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Textbook Question

87. Explain why or why not

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


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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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.

a. For what values of p does this series converge?

Textbook Question

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


a. n!n! = (2n)! for all positive integers n.

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