Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 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 11.2.11
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (kx)ᵏ
Verified step by step guidance1
Identify the given power series: \( \sum_{k=1}^{\infty} (kx)^k \). This can be rewritten as \( \sum_{k=1}^{\infty} k^k x^k \).
To find the radius of convergence, use the root test which involves the limit \( L = \lim_{k \to \infty} \sqrt[k]{|a_k|} \), where \( a_k = k^k x^k \).
Calculate \( \sqrt[k]{|a_k|} = \sqrt[k]{k^k |x|^k} = k |x| \).
Find the limit \( L = \lim_{k \to \infty} k |x| \). For the series to converge, this limit must be less than 1, so set \( \lim_{k \to \infty} k |x| < 1 \).
Since \( \lim_{k \to \infty} k |x| = \infty \) for any \( x \neq 0 \), the radius of convergence is 0, meaning the series converges only at \( x = 0 \). The interval of convergence is therefore \( \{0\} \).
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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 the structure of power series is essential to analyze their convergence behavior depending on the variable x.
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Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center c within which the power series converges absolutely. It can be found using tests like the Ratio Test or Root Test, and it defines the interval where the series behaves well.
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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 all points within the radius of convergence and requires checking endpoints separately to determine if the series converges there.
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08:44Interval of Convergence
Related Practice
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
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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Textbook Question
Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).
Textbook Question
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
√e