Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x sin(1/x)
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 11.2.35
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!
Verified step by step guidance1
Identify the given power series: \(\sum_{k=0}^{\infty} \frac{k^{20} x^{k}}{(2k+1)!}\). We want to find its radius and interval of convergence with respect to \(x\).
Use the Ratio Test to determine the radius of convergence. Consider the ratio of the absolute values of consecutive terms: \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\), where \(a_k = \frac{k^{20} x^{k}}{(2k+1)!}\).
Write the ratio explicitly: \(\left| \frac{(k+1)^{20} x^{k+1} / (2(k+1)+1)!}{k^{20} x^{k} / (2k+1)!} \right| = |x| \cdot \frac{(k+1)^{20}}{k^{20}} \cdot \frac{(2k+1)!}{(2k+3)!}\).
Simplify the factorial expression: \(\frac{(2k+1)!}{(2k+3)!} = \frac{1}{(2k+2)(2k+3)}\). Then analyze the limit as \(k \to \infty\) of the entire expression to find the radius of convergence \(R\).
Once the radius \(R\) is found, determine the interval of convergence by testing the endpoints \(x = -R\) and \(x = R\) separately to check if the series converges or diverges at those points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series
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 power series involves recognizing how the variable x affects convergence depending on its distance from the center.
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Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio or Root Test, which analyze the behavior of the series' terms as k approaches infinity.
Recommended video:
07:36Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It includes all points within the radius of convergence and requires separate checking of endpoints to determine if the series converges there.
Recommended video:
08:44Interval of Convergence
Related Practice
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
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = eˣ
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
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?
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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