Textbook Question
Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
lim ₓ→₀ (eᵃˣ − 1)/x
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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 11.2.17
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (xᵏ/kᵏ)
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Identify the given power series: \(\displaystyle \sum_{k=1}^{\infty} \frac{x^{k}}{k^{k}}\).
To find the radius of convergence, apply the Root Test, which involves computing \(\lim_{k \to \infty} \sqrt[k]{\left| \frac{x^{k}}{k^{k}} \right|} = \lim_{k \to \infty} \frac{|x|}{k}\).
Evaluate the limit: since \(\lim_{k \to \infty} \frac{|x|}{k} = 0\) for all real \(x\), the Root Test tells us the series converges for all \(x\).
Conclude that the radius of convergence \(R\) is infinite, meaning \(R = \infty\).
Since the radius of convergence is infinite, 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_k(x - c)^k, where a_k are coefficients and c is the center. Understanding power series involves recognizing how the variable x affects convergence depending on its distance from the center.
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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 root or ratio test, and it defines the interval where the series behaves well.
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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 separate checking of endpoints to determine if they are included.
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08:44Interval of Convergence
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