Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ
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 11.2.28
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)
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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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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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05:35Ratio Test
Related Practice
Textbook Question
Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?
Textbook Question
{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].
a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.
b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:
q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)
Show that q(x) = −(4/π²)x² + (4/π)x.
c. Graph f, p₂, and q on [0, π].
d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.
e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.
Textbook Question
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
f(x) = (1 + x²)⁻²/³
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.
ln (1 + x²)
Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯
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