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 + x⁴)⁻¹
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 11.2.61
Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
f(x) = ln √(4 − x²)
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Rewrite the function to a more convenient form: \( f(x) = \ln \sqrt{4 - x^2} = \frac{1}{2} \ln(4 - x^2) \). This simplifies the problem to finding the power series for \( \ln(4 - x^2) \).
Express \( \ln(4 - x^2) \) in terms of \( \ln 4 \) and a logarithm of a form suitable for a known power series: \( \ln(4 - x^2) = \ln 4 + \ln\left(1 - \frac{x^2}{4}\right) \).
Recall the known power series for \( \ln(1 - u) = -\sum_{n=1}^\infty \frac{u^n}{n} \) valid for \( |u| < 1 \). Here, set \( u = \frac{x^2}{4} \).
Substitute \( u = \frac{x^2}{4} \) into the series to get \( \ln\left(1 - \frac{x^2}{4}\right) = -\sum_{n=1}^\infty \frac{1}{n} \left(\frac{x^2}{4}\right)^n = -\sum_{n=1}^\infty \frac{x^{2n}}{n 4^n} \).
Combine all parts to write the power series for \( f(x) \): \( f(x) = \frac{1}{2} \ln 4 - \frac{1}{2} \sum_{n=1}^\infty \frac{x^{2n}}{n 4^n} \). The interval of convergence comes from \( |u| = \left|\frac{x^2}{4}\right| < 1 \), which simplifies to \( |x| < 2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series Representation
A power series is an infinite sum of terms in the form a_n(x - c)^n, where c is the center of the series. Representing functions as power series allows approximation and analysis using polynomials. Finding a power series for a function often involves manipulating known series or using derivatives and integrals.
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Intro to Power Series
Known Power Series and Manipulation
Common functions like ln(1+x), 1/(1-x), and sqrt(1-x) have established power series expansions. To find the series for a related function, express it in terms of these known forms and apply algebraic operations, substitutions, or differentiation/integration to derive the new series.
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Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges to the function. It depends on the radius of convergence, often found using the ratio or root test. Determining this interval is crucial to ensure the series accurately represents the function within that domain.
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Related Practice
Textbook Question
Shifting power series If the power series f(x)=∑ cₖ xᵏ has an interval of convergence of |x|<R, what is the interval of convergence of the power series for f(x−a), where a ≠ 0 is a real number?
Textbook Question
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ
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Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ sinᵏ(1/k) xᵏ
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Textbook Question
{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?
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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) = (1 − x)⁻¹
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