Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=3ˣ, 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 11.3.17c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e²ˣ, a = 0
Verified step by step guidance1
Identify the given function and the center of the Taylor series expansion. Here, the function is \(f(x) = e^{2x}\) and the center is \(a = 0\).
Recall the Taylor series expansion for \(e^x\) centered at 0, which is \(\sum_{n=0}^{\infty} \frac{x^n}{n!}\). For \(e^{2x}\), replace \(x\) by \$2x$ to get \(\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}\).
Write the general term of the series as \(a_n = \frac{(2x)^n}{n!}\). To find the interval of convergence, apply the Ratio Test, which involves computing the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Calculate the ratio inside the limit: \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(2x)^{n+1} / (n+1)!}{(2x)^n / n!} \right| = \left| \frac{2x}{n+1} \right|\). Then, find the limit as \(n\) approaches infinity.
Since \(\lim_{n \to \infty} \left| \frac{2x}{n+1} \right| = 0\) for all real \(x\), the Ratio Test tells us the series converges for all real numbers. Therefore, 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.
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (a). For f(x) = e^(2x) at a = 0, the series is formed by evaluating derivatives at 0 and expressing f(x) as a power series in x.
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Taylor Series
Radius and Interval of Convergence
The radius of convergence defines the distance from the center within which the Taylor series converges to the function. The interval of convergence is the set of x-values for which the series converges, determined by tests like the Ratio Test, and may include or exclude endpoints.
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Ratio Test for Convergence
The Ratio Test is used to find the radius of convergence by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges; if greater, it diverges. This test helps identify the interval where the Taylor series for e^(2x) converges.
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Related Practice
Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=1
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Textbook Question
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Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = ln (x − 2), a = 3
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ∑ₖ₌₀∞ (ln 2)ᵏ/k! = 2