Textbook Question
Error Estimates
The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.
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 10.4.17
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 / (2√n + ³√n)
Verified step by step guidance1
Identify the general term of the series: \(a_n = \frac{1}{2\sqrt{n} + \sqrt[3]{n}}\).
Analyze the behavior of the denominator for large \(n\). Since \(\sqrt{n} = n^{1/2}\) and \(\sqrt[3]{n} = n^{1/3}\), the term \(2\sqrt{n}\) grows faster than \(\sqrt[3]{n}\) as \(n\) increases.
Approximate the general term for large \(n\) by focusing on the dominant term in the denominator: \(a_n \approx \frac{1}{2n^{1/2}}\).
Compare the series to a known benchmark series, such as the \(p\)-series \(\sum \frac{1}{n^p}\), where convergence depends on \(p\). Here, the comparison is with \(\sum \frac{1}{n^{1/2}}\).
Recall that the \(p\)-series \(\sum \frac{1}{n^p}\) converges if and only if \(p > 1\). Since \(1/2 < 1\), the comparison series diverges, so by the Comparison Test, the original series also diverges.
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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
Comparison Test
The Comparison Test helps determine convergence by comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test is useful when terms are positive and can be bounded.
Recommended video:
09:25Direct Comparison Test
Asymptotic Behavior of Terms
Analyzing the dominant behavior of terms for large n helps simplify complex expressions. For example, in 1/(2√n + ³√n), the term with the slower decay rate (smaller exponent) dominates. Understanding which term controls the behavior is key to applying convergence tests effectively.
Recommended video:
5:50Asymptotes of Hyperbolas
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 = 2 + (0.1)ⁿ
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Textbook Question
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
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Textbook Question
In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]
Textbook Question
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]
Textbook Question
Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if
b. a = 13/2