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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 11.3.17b
Chapter 11, Problem 11.3.17b

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = e²ˣ, a = 0

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Recall the Taylor series expansion of a function \( f(x) \) about \( a = 0 \) (Maclaurin series) is given by: \[ \displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
Identify the function \( f(x) = e^{2x} \). The derivatives of \( f(x) \) are all of the form \( f^{(n)}(x) = 2^n e^{2x} \). Evaluating at \( x = 0 \), we get: \[ f^{(n)}(0) = 2^n e^0 = 2^n \]
Substitute \( f^{(n)}(0) = 2^n \) into the Taylor series formula: \[ \displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n \]
Rewrite the power series in summation notation explicitly as: \[ \displaystyle e^{2x} = \sum_{n=0}^{\infty} \frac{(2x)^n}{n!} \]
This is the power series representation of \( e^{2x} \) centered at \( a = 0 \). The interval of convergence for this series is all real numbers \( (-\infty, \infty) \) because the exponential function's power series converges everywhere.

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Key Concepts

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

Taylor Series Expansion

The Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (a). For f(x) = e^(2x) at a = 0, the series uses derivatives evaluated at 0 to express f(x) as a power series in (x - 0).
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Taylor Series

Power Series and Summation Notation

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

The interval of convergence is the set of x-values for which the power series converges to the function. Determining this interval involves testing the radius within which the infinite series converges, ensuring 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

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = ln (x − 2), a = 3

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) = ln (1 + x) ≈ x − x²/2

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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

{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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