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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.3.27b
Chapter 11, Problem 11.3.27b

Taylor series


b. Write the power series using summation notation.


f(x)=sin x, a = π/2

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Recall the Taylor series formula for a function \( f(x) \) centered at \( a \): \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \] where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).
Identify the function and center: here, \( f(x) = \sin x \) and \( a = \frac{\pi}{2} \). We need to find the derivatives of \( \sin x \) evaluated at \( x = \frac{\pi}{2} \).
Calculate the first few derivatives of \( \sin x \) and evaluate them at \( x = \frac{\pi}{2} \): - \( f(x) = \sin x \) - \( f'(x) = \cos x \) - \( f''(x) = -\sin x \) - \( f^{(3)}(x) = -\cos x \) - \( f^{(4)}(x) = \sin x \) Evaluate each at \( x = \frac{\pi}{2} \): - \( f\left(\frac{\pi}{2}\right) = 1 \) - \( f'\left(\frac{\pi}{2}\right) = 0 \) - \( f''\left(\frac{\pi}{2}\right) = -1 \) - \( f^{(3)}\left(\frac{\pi}{2}\right) = 0 \) - \( f^{(4)}\left(\frac{\pi}{2}\right) = 1 \)
Notice the pattern in the derivatives evaluated at \( a = \frac{\pi}{2} \): the values cycle through \( 1, 0, -1, 0, 1, \ldots \). This pattern will help simplify the summation.
Write the Taylor series in summation notation using the pattern found: \[ \sin x = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n \] Substitute the values of the derivatives into the summation to express the power series explicitly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a point a, it is expressed as f(x) = Σ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ, where n! is factorial and f⁽ⁿ⁾(a) is the nth derivative at a.
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Derivatives of sin(x)

The derivatives of sin(x) follow a repeating cycle every four derivatives: sin(x), cos(x), -sin(x), -cos(x), then back to sin(x). Evaluating these derivatives at x = π/2 simplifies the coefficients in the Taylor series, as sin(π/2) = 1 and cos(π/2) = 0.
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Derivative of the Natural Exponential Function (e^x)

Summation Notation for Power Series

Summation notation (Σ) compactly expresses infinite series by indicating the general term and the index of summation. Writing the Taylor series in summation form involves identifying the pattern of coefficients and powers of (x - a), allowing a concise representation of the entire series.
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Related Practice
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

b. Find the radius and interval of convergence of the power series for J₀.

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Textbook Question

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = ln (x − 2), a = 3

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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) = sin ⁻¹ x ≈ x

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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

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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

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.

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