Skip to main content
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 10.6.39
Chapter 10, Problem 10.6.39

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.


∑ (k = 1 to ∞) (−1)ᵏ / k⁵

Verified step by step guidance
1
Recognize that the series given is an alternating series of the form \(\sum_{k=1}^\infty \frac{(-1)^k}{k^5}\), where the terms decrease in absolute value and approach zero as \(k\) increases, which means the series converges by the Alternating Series Test.
To estimate the sum with an absolute error less than \(10^{-3}\), use the Alternating Series Estimation Theorem, which states that the absolute error when approximating the sum by the first \(n\) terms is less than or equal to the absolute value of the first omitted term, i.e., \(|R_n| \leq |a_{n+1}|\).
Find the smallest integer \(n\) such that the absolute value of the \((n+1)\)-th term satisfies \(\left| \frac{(-1)^{n+1}}{(n+1)^5} \right| = \frac{1}{(n+1)^5} < 10^{-3}\).
Once \(n\) is found, compute the partial sum \(S_n = \sum_{k=1}^n \frac{(-1)^k}{k^5}\) by adding the first \(n\) terms of the series.
This partial sum \(S_n\) will be an estimate of the infinite series with an absolute error less than \(10^{-3}\), as guaranteed by the Alternating Series Estimation Theorem.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 alternating series like this one, the Alternating Series Test helps determine convergence by checking if terms decrease in absolute value and approach zero.
Recommended video:
06:52
Convergence of an Infinite Series

Alternating Series Estimation Theorem

This theorem states that the absolute error when approximating an alternating series by its partial sum is less than or equal to the absolute value of the first omitted term. It allows us to estimate how many terms are needed to achieve a desired error bound.
Recommended video:
06:00
Geometric Series

Absolute Error and Partial Sums

Absolute error measures the difference between the true sum and the partial sum approximation. By calculating partial sums and comparing the size of the next term, we can ensure the error is below a specified threshold, such as 10⁻³ in this problem.
Recommended video:
04:57
Determining Error and Relative Error
Related Practice
Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (k² / 4ᵏ)

Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (2k + 1)! / (k!)²

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from j = 1 to ∞) 5 / (j² + 4)

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


61. ∑ (k = 1 to ∞) ln((k + 1) / k)

1
views
Textbook Question

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

Textbook Question

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))