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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 10.1.99
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 guidance
1
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.

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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).
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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.
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Definition 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).
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Properties of Functions
Related Practice
Textbook Question

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 2 + (-1)ⁿ

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Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) (1 + 1/n)ⁿ

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 (ⁿ√10)]

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Textbook Question

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 0 to ∞) e^(−2n)

Textbook Question

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

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Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]