Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]
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 10.4.11
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))
Verified step by step guidance1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \). We want to determine if it converges or diverges using the Limit Comparison Test.
Choose a comparison series \( b_n \) that resembles the behavior of \( a_n = \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \) for large \( n \). Simplify the dominant terms: numerator behaves like \( n^2 \), denominator behaves like \( n^3 \), so \( a_n \) behaves like \( \frac{n^2}{n^3} = \frac{1}{n} \). Thus, choose \( b_n = \frac{1}{n} \).
Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n(n + 1)}{(n^2 + 1)(n - 1)}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n(n + 1) \cdot n}{(n^2 + 1)(n - 1)} \).
Simplify the expression inside the limit and evaluate \( L \). If \( L \) is a finite positive number (i.e., \( 0 < L < \infty \)), then \( a_n \) and \( b_n \) have the same behavior regarding convergence.
Since \( \sum \frac{1}{n} \) (the harmonic series) diverges, use the result of the Limit Comparison Test to conclude whether the original series converges or diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit Comparison Test
The Limit Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of the nth terms of the two series. If the limit is a positive finite number, both series either converge or diverge together.
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Limit Comparison Test
Behavior of Rational Functions for Large n
When analyzing series with rational expressions, it is important to understand the dominant terms as n approaches infinity. Simplifying the highest degree terms in numerator and denominator helps approximate the general term's behavior, which is crucial for choosing an appropriate comparison series.
Recommended video:
6:04Intro to Rational Functions
Convergence of p-Series
A p-series is a series of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing or approximating a given series to a p-series helps in applying the Limit Comparison Test effectively, as p-series have well-known convergence properties.
Recommended video:
04:30P-Series and Harmonic Series
Related Practice
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 / √n ]
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 ∞) (n / (3n + 1))ⁿ
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Textbook Question
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
Textbook Question
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
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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 ∞) (tanh n) / n²