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 = 8^(1/n)
1
views
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.99
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (1/n) ∫₁ⁿ (1/x) dx
Verified step by step guidance1
First, recognize that the sequence is defined as \(a_n = \frac{1}{n} \int_1^n \frac{1}{x} \, dx\). Our goal is to analyze the behavior of \(a_n\) as \(n \to \infty\) to determine if it converges or diverges.
Evaluate the integral \(\int_1^n \frac{1}{x} \, dx\). Recall that the integral of \(\frac{1}{x}\) with respect to \(x\) is \(\ln|x|\), so this integral becomes \(\ln n - \ln 1\).
Simplify the integral result using the property of logarithms: \(\ln n - \ln 1 = \ln n\) since \(\ln 1 = 0\).
Substitute the integral back into the sequence definition to get \(a_n = \frac{1}{n} \cdot \ln n\).
Analyze the limit \(\lim_{n \to \infty} \frac{\ln n}{n}\). Since the logarithm grows slower than any linear function, this limit approaches zero, indicating that the sequence converges to 0.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence and Limit
A sequence is an ordered list of numbers defined by a formula for its nth term. The limit of a sequence is the value it approaches as n becomes very large. Understanding how to evaluate limits is essential to determine if a sequence converges (approaches a finite value) or diverges (does not approach a finite value).
Recommended video:
Guided course
8:22
Introduction to Sequences
Definite Integral and its Interpretation
A definite integral ∫₁ⁿ (1/x) dx represents the area under the curve y = 1/x from 1 to n. It can be evaluated using the natural logarithm function, since ∫ (1/x) dx = ln|x| + C. Recognizing this helps simplify the sequence term and analyze its behavior as n grows.
Recommended video:
05:43Definition of the Definite Integral
Properties of the Natural Logarithm Function
The natural logarithm function ln(x) grows slowly and is defined for x > 0. As n increases, ln(n) increases without bound but at a decreasing rate. Understanding how ln(n) behaves relative to n is crucial for evaluating the limit of the sequence aₙ = (1/n) * ln(n).
Recommended video:
06:21Properties of Functions
Related Practice
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 = (2n + 2)! / (2n − 1)!
1
views
Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]