Skip to main content
Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.6.54
Chapter 10, Problem 10.6.54

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]

Verified step by step guidance
1
Recognize that the given series is an alternating series of the form \(\sum_{n=1}^\infty (-1)^{n+1} a_n\) where \(a_n = \frac{n}{n^2 + 1}\). Since the terms alternate in sign, we can consider using the Alternating Series Estimation Theorem to bound the error.
Recall the Alternating Series Estimation Theorem 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., \(|S - S_N| \leq a_{N+1}\).
To ensure the error is less than 0.001, set up the inequality \(a_{N+1} = \frac{N+1}{(N+1)^2 + 1} < 0.001\) and solve for \(N\).
Rewrite the inequality as \(\frac{N+1}{(N+1)^2 + 1} < 0.001\) and multiply both sides by \(((N+1)^2 + 1)\) to get \((N+1) < 0.001 \times ((N+1)^2 + 1)\).
Solve this inequality for \(N\) to find the smallest integer \(N\) such that the error bound is satisfied. This \(N\) will be the number of terms needed to estimate the sum with an error less than 0.001.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Alternating Series

An alternating series is a series whose terms alternate in sign, typically involving (-1)^n or (-1)^(n+1). Such series often converge under specific conditions, and their behavior is key to estimating sums and errors.
Recommended video:
06:00
Geometric Series

Alternating Series Estimation Theorem

This theorem states that the error in approximating the sum of an alternating series by its first n terms is less than or equal to the absolute value of the (n+1)th term. It allows us to determine how many terms are needed to achieve a desired accuracy.
Recommended video:
06:00
Geometric Series

Convergence and Term Behavior

For an alternating series to converge, the absolute value of its terms must decrease monotonically to zero. Understanding the behavior of the term n/(n² + 1) helps verify convergence and apply error bounds effectively.
Recommended video:
05:44
Divergence Test (nth Term Test)
Related Practice
Textbook Question

Error Estimates

The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.

Textbook Question

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

1
views
Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 2 + (0.1)ⁿ

1
views
Textbook Question

Convergence or Divergence

Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

1
views
Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) 1 / (2√n + ³√n)

1
views
Textbook Question

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]