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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 11.3.13b
Chapter 11, Problem 11.3.13b

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x)=2/(1−x)³, a=0

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1
Recognize that the function \( f(x) = \frac{2}{(1 - x)^3} \) can be expressed as a power series centered at \( a = 0 \) by using the generalized binomial series or by differentiating a known geometric series.
Recall the geometric series formula: \( \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \) for \( |x| < 1 \).
Differentiate the geometric series term-by-term to find the series for \( \frac{1}{(1 - x)^2} \), since \( \frac{d}{dx} \left( \frac{1}{1 - x} \right) = \frac{1}{(1 - x)^2} \). This gives \( \frac{1}{(1 - x)^2} = \sum_{n=0}^{\infty} (n+1) x^n \).
Differentiate once more to find the series for \( \frac{1}{(1 - x)^3} \), since \( \frac{d}{dx} \left( \frac{1}{(1 - x)^2} \right) = \frac{2}{(1 - x)^3} \). This yields \( \frac{1}{(1 - x)^3} = \frac{1}{2} \sum_{n=0}^{\infty} (n+1)(n+2) x^n \).
Multiply the series by 2 to get \( f(x) = \frac{2}{(1 - x)^3} = \sum_{n=0}^{\infty} (n+1)(n+2) x^n \), which is the power series representation in summation notation with interval of convergence \( |x| < 1 \).

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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 = 0, the series is expressed as the sum of f⁽ⁿ⁾(0)/n! times xⁿ. This allows complex functions to be approximated by polynomials.
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Taylor Series

Power Series and Summation Notation

A power series is an infinite series of the form Σ cₙ(x - a)ⁿ, where cₙ are coefficients and a is the center. Summation notation compactly expresses this infinite sum, making it easier to manipulate and analyze the series representation of functions.
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Intro to Power Series

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to a finite value. Determining this interval involves testing the radius of convergence, often using the ratio or root test, ensuring the series accurately represents the function within that range.
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Interval of Convergence
Related Practice
Textbook Question

Taylor series


b. Write the power series using summation notation.


f(x) = 1/x, a = 1

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Textbook Question

Symmetry

b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin x.

Textbook Question

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = (1 + x²)⁻¹, a = 0

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

b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

Textbook Question

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


a. Estimate f(0.1) and give a bound on the error in the approximation.


f(x) = tan⁻¹ x ≈ x

Textbook Question

Taylor series


b. Write the power series using summation notation.


f(x) = ln x, a = 3