Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=1
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.1.65c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.
Verified step by step guidance1
Recall that the nth-order Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin polynomial) is given by:
\[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k,\]
where \(f^{(k)}(0)\) is the \(k\)th derivative of \(f\) evaluated at 0.
Consider the function \(f(x) = \sqrt{1 + x^2}\). Notice that \(f(x)\) is an even function because \(f(-x) = \sqrt{1 + (-x)^2} = \sqrt{1 + x^2} = f(x)\).
Since \(f(x)\) is even, its Taylor series expansion around 0 will contain only even powers of \(x\). This is a general property: the Taylor series of an even function centered at 0 contains only even powers, and the Taylor series of an odd function contains only odd powers.
To confirm this, you can compute the first few derivatives of \(f(x)\) at 0 and observe that all derivatives of odd order vanish at 0, i.e., \(f^{(1)}(0) = 0\), \(f^{(3)}(0) = 0\), etc., which means the coefficients of odd powers are zero.
Therefore, the nth-order Taylor polynomial for \(f(x) = \sqrt{1 + x^2}\) centered at 0 contains only even powers of \(x\), making the statement true.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Polynomial
A Taylor polynomial approximates a function near a specific point using derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, expressed as powers of (x - a), where a is the center. It provides a polynomial approximation that matches the function's behavior locally.
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Taylor Polynomials
Even and Odd Functions
An even function satisfies f(-x) = f(x), resulting in Taylor expansions with only even powers of x. An odd function satisfies f(-x) = -f(x), leading to only odd powers in its expansion. Understanding the symmetry of a function helps predict which powers appear in its Taylor series.
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06:21Properties of Functions
Derivatives of Composite Functions
Calculating Taylor polynomials often requires finding derivatives of composite functions like f(x) = √(1 + x²). Using the chain rule and recognizing patterns in derivatives helps determine which terms vanish or remain, influencing the presence of even or odd powers in the polynomial.
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3:48Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).
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Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = cosh 3x, a = 0
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Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
c. Differentiate J₀ twice and show (by keeping terms through x⁶) that J₀ satisfies the equation x² y′′(x) + xy′(x) + x²y(x)=0.
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Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=2/(1−x)³, a=0
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Textbook Question
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).
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