Textbook Question
Find the sum of each series in Exercises 45–52.
∑ (from n = 1 to ∞) [ (2n + 1) / (n²(n + 1)²) ]
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 10.1.123
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
Verified step by step guidance1
First, write down the given sequence explicitly: \(a_n = \frac{2^n 3^n}{n!} = \frac{(2 \cdot 3)^n}{n!} = \frac{6^n}{n!}\).
To determine if the sequence is monotonic, consider the ratio \(\frac{a_{n+1}}{a_n}\) and analyze whether it is greater than, less than, or equal to 1. Compute this ratio as \(\frac{a_{n+1}}{a_n} = \frac{6^{n+1} / (n+1)!}{6^n / n!} = \frac{6^{n+1}}{6^n} \cdot \frac{n!}{(n+1)!} = \frac{6}{n+1}\).
Analyze the ratio \(\frac{6}{n+1}\): for values of \(n\) where \(\frac{6}{n+1} > 1\), the sequence is increasing; where \(\frac{6}{n+1} < 1\), the sequence is decreasing. This helps identify intervals of monotonicity.
To check if the sequence is bounded, consider the behavior of \(a_n = \frac{6^n}{n!}\) as \(n\) becomes very large. Recall that factorial growth (\(n!\)) eventually outpaces exponential growth (\$6^n\(), which suggests the sequence might approach zero for large \)n$.
Summarize the findings: use the ratio test to determine monotonicity intervals and the growth comparison between \$6^n\( and \)n!$ to conclude about boundedness.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Monotonic Sequences
A sequence is monotonic if it is either entirely non-increasing or non-decreasing. To determine monotonicity, compare consecutive terms (aₙ and aₙ₊₁) to see if the sequence consistently increases or decreases. This helps understand the behavior and trend of the sequence over its domain.
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Introduction to Sequences
Bounded Sequences
A sequence is bounded if there exists a real number M such that all terms of the sequence lie within the interval [-M, M]. Checking boundedness involves finding upper and lower limits that the sequence does not exceed, which is crucial for understanding convergence and stability.
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Factorials and Growth Rates
Factorials (n!) grow faster than exponential functions like 2ⁿ or 3ⁿ as n becomes large. Understanding the relative growth rates of factorials versus exponentials is essential to analyze the behavior of the sequence aₙ = (2ⁿ 3ⁿ) / n!, especially when determining limits, monotonicity, and boundedness.
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Related Practice
Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = cos(2x + π/2),a = π/4
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (1 − n) / n2ⁿ
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Textbook Question
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 = 1 to ∞) 1 / n⁰·²
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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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