Textbook Question
Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.
f(x) = e⁻ˣ, a = 0
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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 11.2.21
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! − ⋯
Verified step by step guidance1
First, rewrite the given power series in a general term form. Notice the pattern of powers and factorials: the nth term looks like \((-1)^n \frac{x^{2(n+1)}}{(n+1)!}\), starting from \(n=0\).
Identify the general term \(a_n = (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\). To analyze convergence, it is often easier to express the series as \(\sum_{n=0}^\infty (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\).
Apply the Ratio Test to find the radius of convergence. Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) where \(a_n = (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\).
Simplify the ratio inside the limit: \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{2(n+2)}}{(n+2)!} \cdot \frac{(n+1)!}{x^{2(n+1)}} \right| = \left| \frac{x^2}{n+2} \right|\).
Evaluate the limit as \(n \to \infty\): \(L = \lim_{n \to \infty} \left| \frac{x^2}{n+2} \right| = 0\) for all real \(x\). Since \(L=0 < 1\) for all \(x\), the radius of convergence is infinite, meaning the series converges for all real \(x\). Thus, the interval of convergence is \((-\infty, \infty)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Power Series
A power series is an infinite sum of terms in the form a_n(x - c)^n, where a_n are coefficients and c is the center. Understanding the structure of power series helps analyze their convergence behavior depending on the value of x.
Recommended video:
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 Test or Root Test, indicating the interval where the series represents a valid function.
Recommended video:
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 the radius of convergence and requires checking endpoints separately to determine if the series converges or diverges there.
Recommended video:
08:44Interval of Convergence
Related Practice
Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ/k!
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Textbook Question
Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)
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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
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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)
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Textbook Question
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) = cos x, a = π/4; approximate cos (0.24π)
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