Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) [3ⁿ / n³]
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.5.18
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]
Verified step by step guidance1
Identify the given series: \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \). This is an alternating series because of the factor \( (-1)^n \), which causes the terms to alternate in sign.
Check the absolute value of the terms: \( a_n = n^2 e^{-n} \). Since \( e^{-n} = \frac{1}{e^n} \), the terms involve a polynomial factor \( n^2 \) multiplied by an exponential decay \( e^{-n} \).
Determine if the terms \( a_n = n^2 e^{-n} \) approach zero as \( n \to \infty \). Because exponential decay dominates polynomial growth, \( a_n \to 0 \). This is a necessary condition for convergence.
Consider the absolute convergence by examining the series \( \sum_{n=1}^{\infty} n^2 e^{-n} \). Since \( e^{-n} \) decays faster than any polynomial grows, this series converges by comparison to a convergent geometric series.
Conclude that since the series converges absolutely (the series of absolute values converges), the original alternating series \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \) converges as well.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Understanding this concept is fundamental to analyzing whether a given series sums to a finite value or not.
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Convergence of an Infinite Series
Alternating Series Test
The Alternating Series Test determines convergence for series whose terms alternate in sign. If the absolute value of the terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (-1)^n.
Recommended video:
10:54Alternating Series Test
Comparison and Limit Comparison Tests
These tests compare a given series to a known benchmark series to determine convergence or divergence. By analyzing the behavior of terms relative to simpler series, one can infer the original series' behavior, especially when exponential or polynomial terms are involved.
Recommended video:
07:45Limit Comparison Test
Related Practice
Textbook Question
Error Estimation
In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.
1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1
Textbook Question
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=2 to ∞) 1 / ln n
(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))
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Textbook Question
Telescoping Series
In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.
∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]
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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 ∞) (2x)ⁿ
Textbook Question
Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.
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