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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 11.3.31a
Chapter 11, Problem 11.3.31a

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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1
Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\): \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\] where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Identify the function and center: here, \(f(x) = \ln x\) and \(a = 3\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=3\).
Calculate the first four derivatives of \(f(x) = \ln x\): - \(f(x) = \ln x\) - \(f'(x) = \frac{1}{x}\) - \(f''(x) = -\frac{1}{x^2}\) - \(f^{(3)}(x) = \frac{2}{x^3}\) - \(f^{(4)}(x) = -\frac{6}{x^4}\)
Evaluate each derivative at \(x = 3\): - \(f(3) = \ln 3\) - \(f'(3) = \frac{1}{3}\) - \(f''(3) = -\frac{1}{9}\) - \(f^{(3)}(3) = \frac{2}{27}\) - \(f^{(4)}(3) = -\frac{6}{81} = -\frac{2}{27}\)
Write the first four nonzero terms of the Taylor series using the formula: \[f(x) \approx f(3) + f'(3)(x-3) + \frac{f''(3)}{2!}(x-3)^2 + \frac{f^{(3)}(3)}{3!}(x-3)^3 + \frac{f^{(4)}(3)}{4!}(x-3)^4,\] substituting the values found for each derivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor 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 a. Each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!. This series approximates the function near the point a.
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Taylor Series

Derivatives of the Natural Logarithm Function

To find the Taylor series of f(x) = ln(x), you need to compute its derivatives at x = a. The first derivative is 1/x, and higher derivatives follow a pattern involving factorials and powers of x. Understanding these derivatives is essential to form the series terms.
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Derivative of the Natural Logarithmic Function

Constructing the First Four Nonzero Terms

After finding the derivatives at a, substitute them into the Taylor series formula to write out the first four nonzero terms. This involves careful calculation of each term’s coefficient and power of (x - a), ensuring the series accurately approximates ln(x) near x = 3.
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The First Derivative Test: Finding Local Extrema Example 3
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 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

Textbook Question

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


a. √(1 + 2x)


A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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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)=2/(1−x)³, a=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) = 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