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) = tan ⁻¹ (x/2), 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 11.3.11a
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) = e⁻ˣ, 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.
Find the first few derivatives of the function \(f(x) = e^{-x}\). Note the pattern of the derivatives:
- \(f(x) = e^{-x}\)
- \(f'(x) = -e^{-x}\)
- \(f''(x) = e^{-x}\)
- \(f^{(3)}(x) = -e^{-x}\)
- \(f^{(4)}(x) = e^{-x}\)
and so on, alternating signs.
Evaluate each derivative at \(x=0\):
- \(f(0) = e^{0} = 1\)
- \(f'(0) = -e^{0} = -1\)
- \(f''(0) = e^{0} = 1\)
- \(f^{(3)}(0) = -e^{0} = -1\)
- \(f^{(4)}(0) = e^{0} = 1\)
Write the first four nonzero terms of the Maclaurin series using the formula:
\[\frac{f(0)}{0!} x^0 + \frac{f'(0)}{1!} x^1 + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3.\]
Substitute the values found in the previous step.
Simplify each term by calculating the factorial in the denominator and write the series terms explicitly, keeping the alternating signs and powers of \(x\) clear.
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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 Definition
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 of Exponential Functions
Understanding how to compute derivatives of exponential functions like e^(-x) is essential. The derivative of e^(-x) involves the chain rule, resulting in factors of -1 raised to the power of the derivative order, which affects the sign and form of each term in the series.
Recommended video:
04:50Derivatives of General Exponential Functions
Interval of Convergence
The interval of convergence is the set of x-values for which the Taylor series converges to the function. For exponential functions, the series often converges for all real numbers, but determining this interval ensures the series accurately represents the function within that domain.
Recommended video:
08:44Interval of Convergence
Related Practice
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) = ln x, a = 3
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Textbook Question
Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)
ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2
Textbook Question
ƒ(x) = eˣ, a = 0; e-0.08
b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.
Textbook Question
Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.
x +x³/3 +x⁵/5 +x⁷/7 + ...
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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) = 2ˣ, a = 1