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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.R.60
Chapter 11, Problem 11.R.60

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series.


sin 20°

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1
Recognize that the problem asks for the first four nonzero terms of the Taylor series expansion of \( \sin 20^\circ \). Since Taylor series are typically expressed in radians, first convert \( 20^\circ \) to radians using \( x = 20^\circ \times \frac{\pi}{180} = \frac{\pi}{9} \).
Recall the Taylor series expansion of \( \sin x \) centered at 0 (Maclaurin series): \[ \sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]
Substitute \( x = \frac{\pi}{9} \) into the series to express \( \sin 20^\circ \) as: \[ \sin \left( \frac{\pi}{9} \right) = \frac{\pi}{9} - \frac{\left( \frac{\pi}{9} \right)^3}{3!} + \frac{\left( \frac{\pi}{9} \right)^5}{5!} - \frac{\left( \frac{\pi}{9} \right)^7}{7!} + \cdots \]
Identify the first four nonzero terms from this expansion, which correspond to the powers \( x^{1}, x^{3}, x^{5}, x^{7} \) with alternating signs as shown.
Optionally, consider other centers for the Taylor series (for example, around \( x = \frac{\pi}{6} \) or \( x = 0 \)) if it simplifies the computation, but the Maclaurin series is the most straightforward approach here.

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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 at a single point (the center). It approximates functions near this center, allowing complex functions like sine to be expressed as polynomials. Choosing the center wisely can simplify calculations and improve convergence.
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Trigonometric Function Approximation

Functions like sine can be approximated using their Taylor series expansions around points such as 0 (Maclaurin series) or other angles. This approach helps estimate values like sin 20° by summing a finite number of terms, providing an accurate approximation without a calculator.
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Introduction to Trigonometric Functions

Choosing the Center of Expansion

Selecting the center (a) for the Taylor series affects the complexity and accuracy of the approximation. For sin 20°, centers like 0 or 30° can be chosen. Expanding around a point close to 20° often yields faster convergence and simpler computations for the first few terms.
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Choosing a Convergence Test
Related Practice
Textbook Question

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lim ₙ → ₄ ln (x - 3)/(x² - 16)

Textbook Question

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ƒ(x) = sinh (-3x), n = 3, a = 0

Textbook Question

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.

ƒ(x) = sin 2x, n = 3, a = 0

Textbook Question

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = 1/(4 + x²), a = 0

Textbook Question

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = cos x, a = π/2

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Textbook Question

Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.

ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞