Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=3ˣ, 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 11.3.15a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = (1 + x²)⁻¹, a = 0
Verified step by step guidance1
Recall that the Maclaurin series is a Taylor series centered at \(a = 0\), and is given by the formula:
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\]
where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Start by finding the function and its derivatives at \(x=0\). The function is \(f(x) = (1 + x^2)^{-1}\). Calculate the first few derivatives \(f'(x)\), \(f''(x)\), \(f^{(3)}(x)\), and \(f^{(4)}(x)\).
Evaluate each derivative at \(x=0\) to find \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f^{(3)}(0)\). These values will be used as coefficients in the series.
Write out the first four nonzero terms of the Maclaurin series using the formula:
\[\frac{f(0)}{0!} x^0 + \frac{f'(0)}{1!} x^1 + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots\]
Include only the terms where the coefficient is nonzero.
Simplify each term to express the series clearly, and verify the pattern of coefficients to confirm the first four nonzero terms of the Taylor series centered at 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor and Maclaurin Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.
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Convergence of Taylor & Maclaurin Series
Derivatives and Term Calculation
To find the Taylor series terms, compute successive derivatives of the function at the center point. Each derivative provides the coefficient for the corresponding term in the series. Identifying the first four nonzero terms requires careful differentiation and evaluation at a = 0.
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05:44Divergence Test (nth Term Test)
Interval of Convergence
The interval of convergence is the range of x-values for which the Taylor series converges to the function. It is determined by testing the series' convergence, often using ratio or root tests. Understanding this interval ensures the series accurately represents the function within that domain.
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08:44Interval of Convergence
Related Practice
Textbook Question
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.
Textbook Question
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = 1/x, a = 1
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Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = ln (x − 2), a = 3
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = e²ˣ, a = 0
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).
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