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) = 1/(1 - x), a=0
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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 11.4.14
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x sin(1/x)
Verified step by step guidance1
Recognize that the limit involves the expression \(x \sin\left(\frac{1}{x}\right)\) as \(x\) approaches infinity, which suggests using the Taylor series expansion of \(\sin z\) around \(z=0\) where \(z = \frac{1}{x}\).
Recall the Taylor series expansion for \(\sin z\) around \(z=0\):
\(\sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots\)
Substitute \(z = \frac{1}{x}\) into the series:
\(\sin\left(\frac{1}{x}\right) = \frac{1}{x} - \frac{1}{6x^3} + \frac{1}{120x^5} - \cdots\)
Multiply the entire series by \(x\):
\(x \sin\left(\frac{1}{x}\right) = x \left( \frac{1}{x} - \frac{1}{6x^3} + \frac{1}{120x^5} - \cdots \right) = 1 - \frac{1}{6x^2} + \frac{1}{120x^4} - \cdots\)
Evaluate the limit as \(x \to \infty\) by observing that all terms with \(x\) in the denominator approach zero, so the limit is the constant term remaining in the expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how functions behave as x approaches infinity helps determine if the function approaches a finite value, infinity, or does not exist.
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Cases Where Limits Do Not Exist
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. It approximates functions near that point, allowing complex expressions like sin(1/x) to be expanded into simpler polynomial terms for limit evaluation.
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08:42Taylor Series
Asymptotic Behavior of Functions
Asymptotic behavior studies how functions behave near specific points or at infinity. By analyzing dominant terms in expansions, one can simplify expressions like x sin(1/x) to find limits, focusing on leading terms that dictate the function's growth or decay.
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5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
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Textbook Question
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = x/(1 + x²)² using f(x) = 1/(1 + x²)
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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∞ (k²⁰ xᵏ)/(2k+1)!
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Textbook Question
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?
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Textbook Question
Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
1/(1 − 2x)
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