Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]
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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 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 10.6.94
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
Verified step by step guidance1
Rewrite the given series to understand its general term: the series is \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n^2} \right) \).
Split the series into two separate series using the linearity of summation: \( \sum_{n=1}^{\infty} \frac{1}{n} - \sum_{n=1}^{\infty} \frac{1}{n^2} \).
Analyze the convergence of each series individually: \( \sum_{n=1}^{\infty} \frac{1}{n} \) is the harmonic series, which is known to diverge, and \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a p-series with \( p=2 > 1 \), which converges.
Since the first series diverges and the second converges, consider the behavior of their difference. The divergence of the harmonic series dominates, so the original series behaves like \( \sum \frac{1}{n} \) for large \( n \).
Conclude that the original series diverges because subtracting a convergent series from a divergent one does not make it converge.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Series and Convergence
A series is the sum of the terms of a sequence. To determine if a series converges, we check if the sum approaches a finite limit as the number of terms grows indefinitely. If the partial sums do not approach a finite value, the series diverges.
Recommended video:
06:52
Convergence of an Infinite Series
Comparison Test for Series
The comparison test helps determine convergence by comparing a given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it converges; if larger than a divergent series, it diverges.
Recommended video:
09:25Direct Comparison Test
Behavior of p-Series
A p-series has the form ∑ 1/n^p. It converges if p > 1 and diverges if p ≤ 1. Recognizing parts of a series as p-series helps in analyzing convergence by applying this well-known criterion.
Recommended video:
04:30P-Series and Harmonic Series
Related Practice
Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (1/n) ∫₁ⁿ (1/x) dx
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Textbook Question
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)
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]
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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