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
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 11.2.20
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ/k!
Verified step by step guidance1
Identify the given power series: \( \sum_{k=0}^{\infty} \frac{(2x)^k}{k!} \). This is a power series centered at 0 with general term \( a_k = \frac{(2x)^k}{k!} \).
To find the radius of convergence, use the Ratio Test. Consider the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). Substitute \( a_k \) and \( a_{k+1} \):
\[ L = \lim_{k \to \infty} \left| \frac{(2x)^{k+1} / (k+1)!}{(2x)^k / k!} \right| = \lim_{k \to \infty} \left| \frac{(2x)^{k+1}}{(k+1)!} \cdot \frac{k!}{(2x)^k} \right| = \lim_{k \to \infty} \left| \frac{2x}{k+1} \right| \]
Evaluate the limit \( L \). Since \( \lim_{k \to \infty} \frac{2|x|}{k+1} = 0 \), the Ratio Test tells us the series converges for all real \( x \).
Because the limit \( L = 0 < 1 \) for all \( x \), the radius of convergence \( R = \infty \). This means the power series converges for every real number \( x \).
Therefore, the interval of convergence is \( (-\infty, \infty) \). No endpoints need to be checked since the series converges everywhere.
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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_k(x - c)^k, where a_k are coefficients and c is the center. Understanding power series helps analyze functions represented as infinite polynomials and is essential for determining convergence properties.
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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 x-values for which the power series converges. It includes all points within 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
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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Textbook Question
{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
e⁰ᐧ²⁵, n=4
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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