Textbook Question
Estimate the value of ∑ (from n=2 to ∞) (1 / (n² + 4)) to within 0.1 of its exact value.
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.31
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
Verified step by step guidance1
Identify the general term of the sequence: \(a_n = 2 + (0.1)^n\).
Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.
Analyze the behavior of the term \((0.1)^n\) as \(n \to \infty\). Since \(0.1\) is a number between 0 and 1, \((0.1)^n\) approaches 0 as \(n\) becomes very large.
Use the limit laws to find the limit of the sequence: \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 2 + (0.1)^n \right) = 2 + \lim_{n \to \infty} (0.1)^n\).
Since \(\lim_{n \to \infty} (0.1)^n = 0\), conclude that \(\lim_{n \to \infty} a_n = 2 + 0 = 2\), so the sequence converges to 2.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence Convergence
A sequence converges if its terms approach a specific finite value as n approaches infinity. This limit is the value the sequence gets arbitrarily close to, indicating stability in the sequence's behavior over time.
Recommended video:
Guided course
8:22
Introduction to Sequences
Limit of a Sequence
The limit of a sequence {aₙ} is the value L that the terms aₙ approach as n becomes very large. If such a limit exists and is finite, the sequence converges to L; otherwise, it diverges.
Recommended video:
Guided course
8:22Introduction to Sequences
Behavior of Exponential Terms with Base Between 0 and 1
For a term like (0.1)ⁿ, where the base is between 0 and 1, the value approaches zero as n increases. This property helps determine the limit of sequences involving such terms.
Recommended video:
4:46Solving Exponential Equations Using Like Bases
Related Practice
Textbook Question
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
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 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
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))]
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