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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.69
Chapter 10, Problem 10.2.69

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.


{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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Identify the given sequence as \( a_n = \frac{75}{n \cdot 99^n} + \frac{5^n \sin n}{8^n} \). We want to find \( \lim_{n \to \infty} a_n \) or determine if it diverges.
Analyze the first term \( \frac{75}{n \cdot 99^n} \): since \( 99^n \) grows exponentially and \( n \) grows linearly, this term approaches zero as \( n \to \infty \).
Analyze the second term \( \frac{5^n \sin n}{8^n} \): rewrite it as \( \sin n \cdot \left( \frac{5}{8} \right)^n \). Since \( \left( \frac{5}{8} \right)^n \) is an exponential decay (because \( \frac{5}{8} < 1 \)) and \( \sin n \) is bounded between -1 and 1, this term also approaches zero as \( n \to \infty \).
Combine the limits of both terms: since both approach zero, the sum \( a_n \) approaches zero as \( n \to \infty \).
Conclude that the sequence converges and its limit is zero.

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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 describes the value that the terms approach as the index n goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges. Techniques often involve analyzing the behavior of individual terms or applying limit laws.
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Introduction to Sequences

Exponential Growth and Decay

Exponential terms like 99ⁿ, 5ⁿ, and 8ⁿ grow or decay rapidly depending on the base. Comparing bases helps determine which terms dominate as n increases. For example, if the denominator grows faster than the numerator, the fraction tends to zero, influencing the sequence's limit.
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Exponential Growth & Decay

Behavior of Oscillatory Functions in Sequences

Functions like sin(n) oscillate between -1 and 1, causing terms to fluctuate. When multiplied by terms that tend to zero, the oscillations become negligible, often leading to convergence. Understanding how oscillatory factors interact with exponential terms is key to analyzing sequence limits.
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Related Practice
Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 0 to ∞)k² · 1.001⁻ᵏ

Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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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 ∞) 1 / (2k − √k)

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²