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 = 0 to ∞) (x + 5)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 10.8.2
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) = sin x,a = 0
Verified step by step guidance1
Identify the function and the point of expansion: here, the function is \(f(x) = \sin x\) and the Taylor polynomials are centered at \(a = 0\).
Recall the general formula for the Taylor polynomial of order \(n\) centered at \(a\):
\[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k,\]
where \(f^{(k)}(a)\) is the \(k\)-th derivative of \(f\) evaluated at \(a\).
Calculate the derivatives of \(f(x) = \sin x\) up to the third order and evaluate each at \(x = 0\):
- \(f(x) = \sin x\),
- \(f'(x) = \cos x\),
- \(f''(x) = -\sin x\),
- \(f'''(x) = -\cos x\).
Then find \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f'''(0)\).
Write out each Taylor polynomial using the derivatives evaluated at 0:
- Order 0: \(T_0(x) = f(0)\),
- Order 1: \(T_1(x) = f(0) + f'(0)x\),
- Order 2: \(T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\),
- Order 3: \(T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3\).
Simplify each polynomial by substituting the values of the derivatives at 0 and factorials, leaving the expressions in terms of powers of \(x\).
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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 its derivatives at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n grows.
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Taylor Polynomials
Derivatives of Sine Function
The sine function has a cyclic pattern in its derivatives: f(x) = sin x, f'(x) = cos x, f''(x) = -sin x, f'''(x) = -cos x, and so on. Knowing these derivatives at a specific point is essential for constructing the Taylor polynomial.
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03:53Derivatives of Sine & Cosine
Evaluating Derivatives at the Expansion Point
To build the Taylor polynomial at a point a, each derivative of the function must be evaluated at a. For a = 0, this means calculating sin(0), cos(0), and their higher derivatives at zero, which simplifies the polynomial coefficients.
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Related Practice
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
1
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Textbook Question
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
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Textbook Question
Finding Taylor Series
Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.
e⁻ˣ/²
Textbook Question
Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]