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 10.5.6
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)]
Verified step by step guidance1
Identify the general term of the series as \(a_n = \frac{3^{n+2}}{\ln(n)}\) for \(n \geq 2\).
Write the Ratio Test limit expression: compute \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Substitute \(a_{n+1}\) and \(a_n\) into the ratio: \(\frac{a_{n+1}}{a_n} = \frac{\frac{3^{(n+1)+2}}{\ln(n+1)}}{\frac{3^{n+2}}{\ln(n)}} = \frac{3^{n+3}}{\ln(n+1)} \cdot \frac{\ln(n)}{3^{n+2}}\).
Simplify the ratio expression: \(\frac{a_{n+1}}{a_n} = 3 \cdot \frac{\ln(n)}{\ln(n+1)}\).
Evaluate the limit \(L = \lim_{n \to \infty} 3 \cdot \frac{\ln(n)}{\ln(n+1)}\) and use the Ratio Test criteria: if \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ratio Test
The Ratio Test is a method to determine the convergence or divergence of an infinite 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.
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Ratio Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence, simplifying the analysis of series with positive and negative terms.
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07:51Choosing a Convergence Test
Behavior of Logarithmic and Exponential Functions in Series
Understanding how logarithmic functions grow slowly compared to exponential functions is crucial when analyzing series terms. In this series, the numerator grows exponentially while the denominator grows logarithmically, affecting the limit in the Ratio Test and the overall convergence behavior.
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06:32Derivatives of General Logarithmic Functions
Related Practice
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
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)
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Textbook Question
Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if
b. a = 13/2
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 = (xⁿ / (2n + 1))^(1/n),x > 0
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Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2
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