Textbook Question
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.2.33
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
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Identify the given sequence: \(a_n = \frac{(n+1)!}{n!}\).
Recall the definition of factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
Simplify the expression by canceling common factorial terms: \(\frac{(n+1)!}{n!} = \frac{(n+1) \times n!}{n!} = n+1\).
Analyze the simplified sequence \(a_n = n+1\) as \(n\) approaches infinity.
Conclude whether the sequence converges or diverges based on the behavior of \(n+1\) as \(n \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Sequence Limit
The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges.
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Guided course
8:22
Introduction to Sequences
Factorials and Their Properties
A factorial, denoted n!, is the product of all positive integers up to n. Understanding how factorials grow and simplify, such as (n+1)! = (n+1) × n!, is essential for manipulating sequences involving factorial expressions.
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Behavior of Rational Expressions Involving Factorials
When evaluating limits of sequences with factorials in numerator and denominator, simplifying the expression often reveals growth rates. Recognizing that (n+1)!/n! simplifies to (n+1) helps determine whether the sequence diverges or converges.
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Related Practice
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31.∑ (k = 1 to ∞) 2^(–3k)
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n³⁄(n⁴ + 1)}
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
Textbook Question
Simplify k! / (k + 2)! for any integer k ≥ 0.
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√((1 + 1 / 2n)ⁿ)}
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