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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.3.19a
Chapter 11, Problem 11.3.19a

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

Verified step by step guidance
1
Recall that the Taylor series of a function \(f(x)\) centered at \(a\) is given by the formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Since the center is \(a = 0\), this is a Maclaurin series. We need to find the first four nonzero terms of the series for \(f(x) = \tan^{-1}\left(\frac{x}{2}\right)\), so we will compute the derivatives of \(f(x)\) at \(x=0\) up to the order that gives four nonzero terms.
Start by computing the first derivative: \[f'(x) = \frac{d}{dx} \tan^{-1}\left(\frac{x}{2}\right) = \frac{1}{1 + \left(\frac{x}{2}\right)^2} \cdot \frac{1}{2} = \frac{1}{2 \left(1 + \frac{x^2}{4}\right)} = \frac{1}{2 + \frac{x^2}{2}}.\] Evaluate \(f'(0)\) to get the coefficient for the linear term.
Next, find the higher order derivatives \(f''(x)\), \(f^{(3)}(x)\), and \(f^{(4)}(x)\), and evaluate each at \(x=0\). Use these values to write the terms \[\frac{f^{(n)}(0)}{n!} x^n\] for \(n=0,1,2,3\) (or until you have four nonzero terms).
Finally, write out the Taylor series expansion up to the fourth nonzero term by summing these terms: \[f(x) \approx f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 + \cdots\] This will give you the first four nonzero terms of the Maclaurin series for \(f(x) = \tan^{-1}\left(\frac{x}{2}\right)\).

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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 function's derivatives at a single point. When centered at zero, it is called a Maclaurin series. Each term involves derivatives evaluated at the center, multiplied by powers of (x - a) and divided by factorials.
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Convergence of Taylor & Maclaurin Series

Derivatives of Inverse Trigonometric Functions

To find the Taylor series of f(x) = arctan(x/2), you need to compute derivatives of arctan and apply the chain rule. The derivatives of arctan(x) follow a known pattern, which helps in finding higher-order terms efficiently.
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Derivatives of Other Inverse Trigonometric Functions

Interval of Convergence

The interval of convergence is the range of x-values for which the Taylor series converges to the function. For arctan(x/2), this depends on the radius of convergence determined by the series' ratio or root test, ensuring the series accurately represents the function within that interval.
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Interval of Convergence
Related Practice
Textbook Question

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.


a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

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

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

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) = e⁻ˣ, a=0