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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 11.2.39
Chapter 11, Problem 11.2.39

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = tan⁻¹(4x), a = 0

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Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by \( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \). Here, \( a = 0 \), so the series is centered at zero (Maclaurin series).
Identify the function: \( f(x) = \tan^{-1}(4x) \). We can use the known Maclaurin series for \( \tan^{-1}(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \) for \( |x| < 1 \).
Substitute \( 4x \) in place of \( x \) in the series to get \( \tan^{-1}(4x) = \sum_{n=0}^\infty (-1)^n \frac{(4x)^{2n+1}}{2n+1} = \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1} x^{2n+1}}{2n+1} \).
Write out the first three nonzero terms by plugging in \( n = 0, 1, 2 \): - For \( n=0 \): \( \frac{4^1 x^1}{1} = 4x \) - For \( n=1 \): \( - \frac{4^3 x^3}{3} = - \frac{64 x^3}{3} \) - For \( n=2 \): \( \frac{4^5 x^5}{5} = \frac{1024 x^5}{5} \).
Finally, write the Taylor series in summation notation as \( \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1}}{2n+1} x^{2n+1} \).

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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. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the center point.
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Derivatives of Inverse Trigonometric Functions

Understanding how to differentiate inverse trigonometric functions like arctan(x) is essential. For example, the derivative of arctan(x) is 1/(1 + x^2). When the function is arctan(4x), the chain rule applies, multiplying by the derivative of 4x, which is 4.
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Summation Notation for Series

Summation notation concisely expresses infinite series using the sigma symbol (∑). It includes an index of summation, limits, and a general term formula. Writing the Taylor series in summation form captures all terms compactly and highlights the pattern in coefficients and powers.
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Related Practice
Textbook Question

Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.

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Textbook Question

Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

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{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.


ln 1.04, n=3

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Textbook Question

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² sin x² dx

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Textbook Question

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.


f(x) = eᶜᵒˢ ˣ

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Textbook Question

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

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