Textbook Question
Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.
f(x) = e⁻ˣ, 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 11.4.55
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)
Verified step by step guidance1
Recognize that the given series is a geometric series of the form \(\sum_{k=0}^\infty ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
Identify the first term \(a\) by substituting \(k=0\) into the series: \(a = \frac{x^0}{2^0} = 1\).
Identify the common ratio \(r\) by looking at the general term: \(r = \frac{x}{2}\).
Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\).
Write the function represented by the power series as \(f(x) = \frac{1}{1 - \frac{x}{2}}\) and simplify if desired.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series Representation
A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding how functions can be expressed as power series allows us to analyze and approximate functions within their radius of convergence.
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Intro to Power Series
Geometric Series and Its Sum
A geometric series is a power series where each term is a constant ratio times the previous term, ∑ r^k. Its sum converges to 1/(1 - r) when |r| < 1. Recognizing a power series as geometric helps identify the function it represents.
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06:00Geometric Series
Radius and Interval of Convergence
The radius of convergence defines the interval around the center where the power series converges to a finite value. For the series ∑ (x^k)/(2^k), convergence depends on |x/2| < 1, which is essential to determine the domain where the function representation is valid.
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07:36Radius of Convergence
Related Practice
Textbook Question
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.
cos 2
Textbook Question
Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.
∫₀1/2 tan⁻¹ x dx
Textbook Question
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
Textbook Question
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)
Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cos⁻¹ x, n = 2, a = 1/2