Textbook Question
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) = 2x/(1 + x²)²
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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 11.2.13
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ sinᵏ(1/k) xᵏ
Verified step by step guidance1
Identify the general term of the power series: \(a_k = \sin^k\left(\frac{1}{k}\right) x^k\).
To find the radius of convergence, use the root test which involves calculating \(\limsup_{k \to \infty} \sqrt[k]{|a_k|} = \limsup_{k \to \infty} \sqrt[k]{\left|\sin^k\left(\frac{1}{k}\right) x^k\right|}\).
Simplify the expression inside the limit: \(\sqrt[k]{|a_k|} = \left|\sin\left(\frac{1}{k}\right)\right| \cdot |x|\).
Evaluate the limit \(\lim_{k \to \infty} \sin\left(\frac{1}{k}\right)\) by using the fact that \(\sin y \approx y\) for small \(y\), so \(\sin\left(\frac{1}{k}\right) \approx \frac{1}{k}\), which tends to 0 as \(k \to \infty\).
Since the limit of \(\sin\left(\frac{1}{k}\right)\) is 0, the root test limit becomes \(0 \cdot |x| = 0\) for all \(x\), indicating the radius of convergence is infinite. Then, verify the interval of convergence by checking the behavior of the series at all real \(x\) values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series and Convergence
A power series is an infinite sum of terms in the form a_k x^k, where a_k depends on k. Understanding convergence means determining for which values of x the series sums to a finite value. This involves analyzing the behavior of the coefficients and the variable x.
Recommended video:
05:58
Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center of the power series (usually zero) within which the series converges absolutely. It can be found using tests like the root or ratio test applied to the coefficients a_k. The radius defines an interval on the x-axis where the series behaves well.
Recommended video:
07:36Radius of Convergence
Interval of Convergence
The interval of convergence includes all x-values for which the power series converges, typically from -R to R, where R is the radius of convergence. Endpoints must be checked separately since convergence there is not guaranteed. This interval determines the domain where the series represents a valid function.
Recommended video:
08:44Interval of Convergence
Related Practice
Textbook Question
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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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?
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Textbook Question
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
∑ₖ₌₀∞ e⁻ᵏˣ
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Textbook Question
Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is
eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞
Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.
f(x) = x²eˣ
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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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