Textbook Question
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 10.3.8
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 2 to ∞) ln(n²) / n
Verified step by step guidance1
First, identify the function corresponding to the terms of the series: define \( f(x) = \frac{\ln(x^2)}{x} \) for \( x \geq 2 \).
Check the conditions for the Integral Test: verify that \( f(x) \) is continuous, positive, and decreasing on the interval \([2, \infty)\).
Simplify the function \( f(x) \) using logarithm properties: \( \ln(x^2) = 2 \ln(x) \), so \( f(x) = \frac{2 \ln(x)}{x} \).
Set up the improper integral to apply the Integral Test: \( \int_2^{\infty} \frac{2 \ln(x)}{x} \, dx \).
Evaluate the integral (or analyze its behavior) to determine if it converges or diverges, which will tell you whether the series converges or diverges by the Integral Test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integral Test for Series Convergence
The Integral Test determines the convergence or divergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing for x ≥ N, then the series and the integral either both converge or both diverge.
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Choosing a Convergence Test
Conditions for Applying the Integral Test
Before using the Integral Test, verify that the function f(x) representing the series terms is positive, continuous, and decreasing on the interval [N, ∞). These conditions ensure the integral comparison is valid and the test can be applied correctly.
Recommended video:
07:25Integral Test
Evaluating the Improper Integral
To apply the Integral Test, evaluate the improper integral of f(x) from N to infinity. This often involves techniques like substitution or integration by parts. The convergence or divergence of this integral directly determines the behavior of the original series.
Recommended video:
11:11Improper Integrals: Infinite Intervals
Related Practice
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
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Textbook Question
In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.
aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ
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Textbook Question
Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)
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 = (n + 3) / (n² + 5n + 6)
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Textbook Question
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.