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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 11.3.25b
Chapter 11, Problem 11.3.25b

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = ln (x − 2), a = 3

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Identify the function and the center of the Taylor series expansion. Here, the function is \(f(x) = \ln(x - 2)\) and the center is \(a = 3\).
Rewrite the function in terms of \((x - a)\) to express it as a power series centered at \(x = 3\). Set \(u = x - 3\), so that \(x - 2 = (x - 3) + 1 = u + 1\).
Recall the Taylor series expansion for \(\ln(1 + u)\) around \(u = 0\), which is \(\ln(1 + u) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{u^n}{n}\) for \(|u| < 1\).
Substitute back \(u = x - 3\) into the series to write \(f(x)\) as a power series centered at \(x = 3\): \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\).
Express the final answer in summation notation: \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\), which represents the power series expansion of \(\ln(x - 2)\) about \(x = 3\).

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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 a. It approximates the function near that point using powers of (x - a), allowing complex functions to be expressed as polynomials.
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Taylor Series

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation concisely expresses this infinite sum, making it easier to manipulate and analyze the series.
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Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function. Determining this interval involves testing the radius of convergence and checking endpoints to ensure the series accurately represents the function within that range.
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Related Practice
Textbook Question

{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ

b. Find the radius and interval of convergence of the power series for J₀.

Textbook Question

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


b. Estimate f(0.2) and give a bound on the error in the approximation.


f(x) = sin x ≈ x

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.


b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.

Textbook Question

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = e²ˣ, a = 0

Textbook Question

{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ

a. Write out the first four terms of J₀.

Textbook Question

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


b. Estimate f(0.2) and give a bound on the error in the approximation.


f(x) = sin ⁻¹ x ≈ x

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