Domanda del libro di testo
Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
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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 11.2.28
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)
Guida verificata passo dopo passo1
Identify the given power series: \( \sum_{k=2}^{\infty} \frac{(x+3)^k}{k \ln^2(k)} \). Notice that the series is centered at \( x = -3 \) because the term is \( (x+3)^k \).
To find the radius of convergence, apply the Root Test or the Ratio Test. Here, the Root Test is convenient. Consider the general term \( a_k = \frac{(x+3)^k}{k \ln^2(k)} \). Compute \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \).
Calculate \( \sqrt[k]{|a_k|} = \sqrt[k]{\frac{|x+3|^k}{k \ln^2(k)}} = |x+3| \cdot \sqrt[k]{\frac{1}{k \ln^2(k)}} \). As \( k \to \infty \), \( \sqrt[k]{k} \to 1 \) and \( \sqrt[k]{\ln^2(k)} \to 1 \), so the limit simplifies to \( |x+3| \).
The Root Test states that the series converges if \( \lim_{k \to \infty} \sqrt[k]{|a_k|} < 1 \), so the radius of convergence \( R \) satisfies \( |x+3| < 1 \). Thus, the radius of convergence is \( R = 1 \).
Next, determine the interval of convergence by checking the endpoints \( x = -3 - 1 = -4 \) and \( x = -3 + 1 = -2 \). Substitute these values into the original series and analyze convergence using appropriate tests (such as the Comparison Test or Integral Test) because the behavior at endpoints depends on the series without the \( (x+3)^k \) factor.
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Radius of Convergence
The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It is found using tests like the Ratio or Root Test, and it defines the interval around the center where the series behaves well.
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07:36
Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It includes the radius of convergence and requires checking endpoints separately, as convergence at these points is not guaranteed by the radius alone.
Video consigliato:
08:44Interval of Convergence
Ratio Test for Convergence
The Ratio Test determines convergence by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges; if greater than one, it diverges. It is especially useful for power series to find the radius of convergence.
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Percorso guidato
05:35Ratio Test
Pratica correlata
Domanda del libro di testo
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)
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Domanda del libro di testo
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ
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Domanda del libro di testo
Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.
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Domanda del libro di testo
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (x/3)ᵏ
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Domanda del libro di testo
Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!
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