Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = cosh 3x, a = 0
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 11.3.67b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.
Verified step by step guidance1
Recall that a Taylor series of a function \(f(x)\) centered at \(a\) is given by the infinite sum \(\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n\), where \(f^{(n)}(a)\) denotes the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
To determine if \(f(x) = \csc x\) has a Taylor series centered at \(x = \frac{\pi}{2}\), first check if \(f\) is differentiable at \(x = \frac{\pi}{2}\) and in some open interval around it.
Evaluate \(f\left(\frac{\pi}{2}\right) = \csc\left(\frac{\pi}{2}\right) = 1\), which is finite, so the function is defined at \(x = \frac{\pi}{2}\).
Next, consider the behavior of \(\csc x\) near \(x = \frac{\pi}{2}\). Since \(\sin x\) is nonzero and smooth around \(\frac{\pi}{2}\), \(\csc x = \frac{1}{\sin x}\) is also smooth and infinitely differentiable there.
Therefore, \(f(x) = \csc x\) is infinitely differentiable at \(x = \frac{\pi}{2}\), so it does have a Taylor series centered at \(\frac{\pi}{2}\). The Taylor series converges to \(f(x)\) in some interval around \(\frac{\pi}{2}\) where \(\sin x \neq 0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series and Center of Expansion
A Taylor series represents a function as an infinite sum of derivatives at a specific point called the center. The series converges to the function near this point if the function is sufficiently smooth and differentiable there. Understanding the center is crucial to determine if the series exists and converges.
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Taylor Series
Domain and Differentiability of csc(x)
The function csc(x) = 1/sin(x) is undefined where sin(x) = 0, causing vertical asymptotes. For a Taylor series to exist at a point, the function must be defined and infinitely differentiable there. Checking if csc(x) is defined and smooth at π/2 is essential.
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5:10Finding the Domain and Range of a Graph
Radius of Convergence and Singularities
The radius of convergence of a Taylor series is limited by the nearest singularity to the center point. Since csc(x) has singularities where sin(x) = 0, the distance from π/2 to these points affects the convergence. Identifying these singularities helps determine if the Taylor series is valid.
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07:36Radius of Convergence
Related Practice
Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e⁻ˣ, a=0
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) = tan x ≈ x
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
f(x) = 1/x², a=1
Textbook Question
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
b. Integrate the series to find a Taylor series for Si.
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) =√(1+x) ≈ 1 + x/2