Textbook Question
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]
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 10.8.10
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
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Identify the function and the point about which the Taylor polynomial is generated: here, \( f(x) = \sqrt{1 - x} \) and \( a = 0 \).
Calculate the derivatives of \( f(x) \) up to the third order: find \( f'(x) \), \( f''(x) \), and \( f'''(x) \).
Evaluate the function and its derivatives at \( x = 0 \): compute \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f'''(0) \).
Write the Taylor polynomials of orders 0, 1, 2, and 3 using the formula: \[ T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k \], substituting \( a = 0 \) and the evaluated derivatives.
Express each polynomial explicitly: the 0th order is just \( f(0) \), the 1st order adds the linear term, the 2nd order adds the quadratic term, and the 3rd order adds the cubic term, all centered at \( x = 0 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Polynomials
Taylor polynomials approximate a function near a point a by using a finite sum of derivatives of the function at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n increases.
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Derivatives of Functions
Derivatives measure the rate of change of a function. To find Taylor polynomials, you must compute the function's derivatives at the point a, as these derivatives determine the coefficients of the polynomial terms.
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Evaluating at the Expansion Point
The Taylor polynomial is centered at a specific point a, where the function and its derivatives are evaluated. This evaluation ensures the polynomial matches the function's value and slope behavior at that point, making the approximation accurate near a.
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Related Practice
Textbook Question
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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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
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 / (n³ 3ⁿ) ]
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Textbook Question
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
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