Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=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 11.3.21c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=3ˣ, a=0
Verified step by step guidance1
Recall that the Taylor series for the function \(f(x) = 3^x\) centered at \(a=0\) (Maclaurin series) can be expressed using the general formula for the Taylor series:
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\]
Find the \(n\)th derivative of \(f(x) = 3^x\). Since \(3^x = e^{x \ln 3}\), the \(n\)th derivative is
\[f^{(n)}(x) = (\ln 3)^n 3^x\]
Therefore, at \(x=0\),
\[f^{(n)}(0) = (\ln 3)^n 3^0 = (\ln 3)^n\]
Substitute the derivatives into the Taylor series formula to get
\[f(x) = \sum_{n=0}^{\infty} \frac{(\ln 3)^n}{n!} x^n\]
To determine the interval of convergence, apply the Ratio Test to the series terms
\[a_n = \frac{(\ln 3)^n}{n!} x^n\]
Calculate
\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(\ln 3)^{n+1}}{(n+1)!} x^{n+1} \cdot \frac{n!}{(\ln 3)^n x^n} \right| = \lim_{n \to \infty} \left| \frac{(\ln 3) x}{n+1} \right|\]
Since
\[\lim_{n \to \infty} \frac{|(\ln 3) x|}{n+1} = 0\]
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) = 3^x at a = 0, the series expresses 3^x as a power series in terms of (x - 0), allowing approximation of the function near zero.
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Taylor Series
Radius and Interval of Convergence
The interval of convergence is the set of x-values for which the Taylor series converges to the function. It is determined by finding the radius of convergence, often using the Ratio or Root Test, and then checking the endpoints to see if the series converges there.
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07:36Radius of Convergence
Ratio Test for Convergence
The Ratio Test is a method to determine the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive.
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05:35Ratio Test
Related Practice
Textbook Question
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e²ˣ, a = 0
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Textbook Question
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
c. Differentiate J₀ twice and show (by keeping terms through x⁶) that J₀ satisfies the equation x² y′′(x) + xy′(x) + x²y(x)=0.
Textbook Question
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
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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
1
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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