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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 11.2.69
Chapter 11, Problem 11.2.69

Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)


 ∑ₖ₌₀∞ e⁻ᵏˣ

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Recognize that the given series is \( \sum_{k=0}^{\infty} e^{-kx} \). This is a geometric series where each term can be written as \( (e^{-x})^k \).
Recall the formula for the sum of an infinite geometric series: \( \sum_{k=0}^{\infty} r^k = \frac{1}{1-r} \), which converges if and only if \( |r| < 1 \).
Identify the common ratio \( r = e^{-x} \) in this series. To apply the formula, we need to find the values of \( x \) such that \( |e^{-x}| < 1 \).
Since \( e^{-x} > 0 \) for all real \( x \), the inequality \( e^{-x} < 1 \) simplifies to \( -x < 0 \), or \( x > 0 \). This gives the interval of convergence.
Write the function represented by the series as \( f(x) = \frac{1}{1 - e^{-x}} \) for \( x > 0 \).

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Key Concepts

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

Geometric Series and Its Sum

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. The sum of an infinite geometric series with first term a and common ratio r (|r| < 1) is a/(1 - r). Recognizing the given series as geometric allows us to express it as a closed-form function.
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Geometric Series

Interval of Convergence

The interval of convergence is the set of values for the variable x for which the infinite series converges. For geometric series, convergence requires the absolute value of the common ratio to be less than one. Determining this interval ensures the function representation is valid within that domain.
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Interval of Convergence

Exponential Functions and Their Properties

Exponential functions of the form e^{kx} have properties that simplify series expressions, especially when combined with geometric series concepts. Understanding how e^{-kx} behaves as k varies helps in identifying the ratio and analyzing convergence.
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Properties of Functions
Related Practice
Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

Textbook Question

Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)


∑ₖ₌₀∞(√x − 2)ᵏ

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Textbook Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


{(eˣ−1)/x if x ≠ 1, 1 if x = 1

Textbook Question

Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is


eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞


Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.


f(x) = x²eˣ

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Textbook Question

Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

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Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ sinᵏ(1/k) xᵏ

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