Textbook Question
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
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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 10.2.61
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = e⁻ⁿcosn
Verified step by step guidance1
Identify the given sequence: \(a_n = e^{-n} \cos n\).
Recall that \(e^{-n}\) represents an exponential decay term, which approaches 0 as \(n\) approaches infinity.
Note that \(\cos n\) oscillates between -1 and 1 for all integer values of \(n\) and does not have a limit.
Since \(e^{-n}\) tends to 0 and \(\cos n\) is bounded, the product \(e^{-n} \cos n\) will be squeezed between \(-e^{-n}\) and \(e^{-n}\).
Apply the Squeeze Theorem: because both \(-e^{-n}\) and \(e^{-n}\) approach 0, the sequence \(a_n = e^{-n} \cos n\) also approaches 0 as \(n\) approaches infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits of Sequences
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 fixed number, the sequence converges; otherwise, it diverges. Understanding limits helps determine the long-term behavior of sequences.
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Guided course
8:22
Introduction to Sequences
Exponential Decay
Exponential decay refers to functions of the form e^(-n), which decrease rapidly towards zero as n increases. In sequences, this factor often dominates the behavior, causing terms to shrink and potentially leading the sequence to converge to zero.
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09:29Exponential Growth & Decay
Oscillatory Behavior of Trigonometric Functions
Functions like cos(n) oscillate between -1 and 1 without settling to a single value. When combined with other factors, such as exponential decay, the oscillations influence the sequence's behavior but may be dampened if multiplied by a term tending to zero.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 3 to ∞) 5 / (2 + lnk)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!
Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)5¹⁻²ᵏ
Textbook Question
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))