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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 11.2.67
Chapter 11, Problem 11.2.67

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

Verified step by step guidance
1
Recognize that the given series is a geometric series of the form \(\sum_{k=0}^\infty r^k\), where the common ratio \(r\) is \(\sqrt{x} - 2\).
Recall that a geometric series \(\sum_{k=0}^\infty r^k\) converges to the function \(\frac{1}{1-r}\) when \(|r| < 1\).
Write the function represented by the series as \(f(x) = \frac{1}{1 - (\sqrt{x} - 2)}\).
Simplify the denominator to get \(f(x) = \frac{1}{1 - \sqrt{x} + 2} = \frac{1}{3 - \sqrt{x}}\).
Determine the interval of convergence by solving the inequality \(|\sqrt{x} - 2| < 1\). This involves considering the values of \(x\) for which the absolute value condition holds, and also ensuring \(x \geq 0\) since \(\sqrt{x}\) is defined for non-negative \(x\).

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Key Concepts

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

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. It converges to a finite value if the absolute value of the ratio is less than 1, and its sum is given by S = a / (1 - r), where a is the first term and r is the ratio.
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Interval of Convergence

The interval of convergence is the set of all x-values for which a given series converges. For power or related series, it is found by applying convergence tests, often involving inequalities on the variable to ensure the series terms approach zero.
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Interval of Convergence

Manipulating Series with Functions

To find the function represented by a series, one must recognize the pattern of terms and rewrite the series in a closed form. This often involves identifying the series type and substituting expressions (like √x − 2) as the variable to simplify and find the sum function.
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Related Practice
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∞ e⁻ᵏˣ

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

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

Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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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)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².

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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