Domanda del libro di testo
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x sin(1/x)
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 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)!
Guida verificata passo dopo passo1
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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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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05:58
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.
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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.
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08:44Interval of Convergence
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
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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