Textbook Question
(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.
a. ∑ (from n=2 to ∞) [1 / (n ln n)]
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 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.PE.18
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
Verified step by step guidance1
Identify the general term of the sequence: \(a_n = \frac{(-4)^n}{n!}\).
Recall that \(n!\) (n factorial) grows much faster than any exponential function \(c^n\) as \(n\) approaches infinity.
To determine convergence, analyze the limit \(\lim_{n \to \infty} \frac{(-4)^n}{n!}\).
Since the factorial in the denominator grows faster than the exponential in the numerator, the terms \(a_n\) approach zero as \(n\) becomes very large.
Conclude that the sequence converges and its limit is \(0\).
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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. Determining convergence involves analyzing the behavior of the nth term and checking if the limit exists and is finite.
Recommended video:
Guided course
8:22
Introduction to Sequences
Factorials and Growth Rates
Factorials (n!) grow much faster than exponential functions like (-4)^n. Understanding the relative growth rates helps in evaluating limits involving factorials and powers, often leading to convergence when factorials dominate.
Recommended video:
5:22Factorials
Limit of a Sequence
The limit of a sequence aₙ is the value that aₙ approaches as n becomes very large. Calculating this limit often involves applying limit laws, comparing growth rates, or using known limits of standard sequences.
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8:22Introduction to Sequences
Related Practice
Textbook Question
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) xⁿ/nⁿ
Textbook Question
The series
sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯
converges to sec x for −π/2 < x < π/2.
a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?
Textbook Question
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (csch n)xⁿ
Textbook Question
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
a. Converges absolutely for x = 1
Textbook Question
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 1 + (0.9)ⁿ
1
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