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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.R.89b
Chapter 10, Problem 10.R.89b

89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.


b. Find an upper bound for the remainder Rₙ.


89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

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1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^5} \). This is a p-series with \( p = 5 \), which converges because \( p > 1 \).
Recall that the remainder \( R_n \) after \( n \) terms is the difference between the infinite sum and the partial sum up to \( n \): \( R_n = S - S_n \). We want to find an upper bound for \( R_n \) when \( n = 5 \).
Use the integral test remainder estimate for a decreasing positive function \( f(k) = \frac{1}{k^5} \). The upper bound for the remainder \( R_n \) is given by the integral from \( n \) to infinity of \( f(x) \): \[ R_n \leq \int_n^{\infty} \frac{1}{x^5} \, dx \]
Evaluate the improper integral: \[ \int_n^{\infty} \frac{1}{x^5} \, dx = \lim_{t \to \infty} \int_n^t x^{-5} \, dx \] Find the antiderivative of \( x^{-5} \), which is \( \frac{x^{-4}}{-4} = -\frac{1}{4x^4} \).
Apply the limits to the antiderivative: \[ \lim_{t \to \infty} \left( -\frac{1}{4t^4} + \frac{1}{4n^4} \right) = 0 + \frac{1}{4n^4} = \frac{1}{4n^4} \] Thus, the upper bound for the remainder \( R_n \) when \( n = 5 \) is \( \frac{1}{4 \times 5^4} \).

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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/k⁵ converges because its terms decrease rapidly and resemble a p-series with p > 1. Understanding convergence ensures that the remainder after n terms is meaningful and finite.
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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 sum. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.
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Integral Test and Remainder Bounds

The integral test can be used to estimate the remainder of a decreasing positive term series. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This provides a practical way to find upper bounds for the error in series approximations.
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Integral Test
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For each convergent series and given value of n, complete the following.


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