Textbook Question
In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!
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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 10.4.35
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
Verified step by step guidance1
First, write down the general term of the series: \(a_n = \frac{1 - n}{n 2^n}\).
Observe the behavior of the terms as \(n\) becomes large. Since \$2^n\( grows exponentially and \)n$ grows linearly, the denominator grows very fast compared to the numerator.
To determine convergence, consider applying the Ratio Test, which is useful for series involving terms with exponential components. The Ratio Test states to compute \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Calculate \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(1 - (n+1)) / ((n+1) 2^{n+1})}{(1 - n) / (n 2^n)} \right|\) and simplify this expression step-by-step.
Evaluate the limit \(L\) as \(n\) approaches infinity. If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive and other methods should be considered.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges.
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06:52
Convergence of an Infinite Series
Ratio Test
The Ratio Test is used to determine the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
Recommended video:
05:35Ratio Test
Behavior of Exponential Terms in Series
Exponential terms like 2^n in the denominator grow very rapidly, often causing terms to approach zero quickly. This rapid decay can lead to convergence, especially when combined with polynomial terms in the numerator or denominator.
Recommended video:
06:00Geometric Series
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
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 = 1 to ∞) [ (x − 1)ⁿ / (n³ 3ⁿ) ]
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Textbook Question
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
Textbook Question
Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))
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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