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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.2.67
Capitolo 11, Problema 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

Guida verificata passo dopo passo
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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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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Geometric Series

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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Geometric Series
Pratica correlata
Domanda del libro di testo

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = 2x/(1 + x²)²

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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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Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

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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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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.

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

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

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