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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 11.3.57
Chapter 11, Problem 11.3.57

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


(1 + 4x)⁻²

Verified step by step guidance
1
Recall the given Maclaurin series for the function \((1 + x)^{-2}\): \[ (1 + x)^{-2} = 1 - 2x + 3x^{2} - 4x^{3} + \cdots \]
To find the Maclaurin series for \((1 + 4x)^{-2}\), use substitution by replacing \(x\) with \$4x$ in the original series. This gives: \[ (1 + 4x)^{-2} = 1 - 2(4x) + 3(4x)^{2} - 4(4x)^{3} + \cdots \]
Simplify each term by calculating the powers and coefficients: - The linear term: \(-2 \times 4x = -8x\) - The quadratic term: \(3 \times (4x)^2 = 3 \times 16x^2 = 48x^2\) - The cubic term: \(-4 \times (4x)^3 = -4 \times 64x^3 = -256x^3\)
Write out the first four nonzero terms explicitly: \[ (1 + 4x)^{-2} = 1 - 8x + 48x^{2} - 256x^{3} + \cdots \]
Confirm the interval of convergence remains valid by considering the substitution: since the original series converges for \(|x| < 1\), the new series converges for \(|4x| < 1\), or equivalently \(|x| < \frac{1}{4}\).

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Key Concepts

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

Maclaurin Series

A Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of terms involving powers of x and derivatives evaluated at zero. Understanding this allows approximation of functions near zero using polynomials.
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Convergence of Taylor & Maclaurin Series

Binomial Series Expansion

The binomial series generalizes the binomial theorem to any real exponent, expressing (1 + x)^n as an infinite power series. For negative integer exponents, it produces alternating terms with coefficients derived from binomial coefficients, useful for expanding functions like (1 + x)^-2.
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Geometric Series

Substitution and Factoring in Power Series

Substitution involves replacing x with another expression inside a known power series to find expansions of related functions. Factoring helps simplify expressions before expansion. Together, they enable finding series for functions like (1 + 4x)^-2 by adapting the known series for (1 + x)^-2.
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Intro to Power Series
Related Practice
Textbook Question

{Use of Tech} Newton's derivation of the sine and arcsine series Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.

a. Referring to the figure, show that x = sin s or s = sin ⁻¹ x.

b. The area of a circular sector of radius r subtended by an angle θ is 1/2r²θ. Show that the area of the circular sector APE is s/2, which implies that

s = 2 ∫₀ˣ √(1 − t²) dt − x √(1 −x²)

c. Use the binomial series for f(x) = √(1 − x²) to obtain the first few terms of the Taylor series for s=sin ⁻¹ x.

d. Newton next inverted the series in part (c) to obtain the Taylor series for x=sin s. He did this by assuming sin s = ∑ aₖ sᵏ and solving x = sin(sin ⁻¹ x) for the coefficients aₖ. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).

Textbook Question

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.


f(x) = ∫₀ˣ sin t² dt

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Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ k(x−1)ᵏ

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Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹

Textbook Question

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

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Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}

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