Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.2.2

Chapter 11, Problem 11.2.2

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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Recognize that a power series is generally expressed in the form \(\sum_{k=0}^\infty c_k (x - a)^k\), where \(a\) is the center of the series and \(c_k\) are the coefficients depending only on \(k\).

Rewrite the given series \(\sum_{k=0}^\infty (5x - 20)^k\) by factoring the expression inside the power to isolate \(x\): note that \(5x - 20 = 5(x - 4)\).

Express the series as \(\sum_{k=0}^\infty [5(x - 4)]^k = \sum_{k=0}^\infty 5^k (x - 4)^k\) to match the standard power series form.

Identify the center \(a\) of the power series as the value of \(x\) that makes the term inside the parentheses zero, which is \(a = 4\).

From the rewritten series, determine the coefficients \(c_k\) as \(c_k = 5^k\), since the series is now \(\sum_{k=0}^\infty c_k (x - 4)^k\).

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Key Concepts

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

Power Series Definition

A power series is an infinite series of the form ∑ₖ₌₀ ∞ cₖ(x - a)ᵏ, where cₖ are coefficients and a is the center. It represents a function as a sum of powers of (x - a), converging within a certain radius. Recognizing the structure helps determine if a given series qualifies as a power series.

Center of a Power Series

The center 'a' of a power series is the value around which the series is expanded, appearing in the term (x - a). Identifying 'a' involves rewriting the series so that the variable part matches (x - a)ᵏ. This center is crucial for understanding the interval of convergence and the behavior of the series.

Coefficients of a Power Series

The coefficients cₖ in a power series determine the weight of each term (x - a)ᵏ. Finding cₖ often requires expressing the series in the standard form and comparing terms. These coefficients are essential for analyzing the function represented and for operations like differentiation or integration of the series.

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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∞(√x − 2)ᵏ

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Textbook Question

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) = ln √(1 − x²)

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Textbook Question

Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

f(3x) = ln (1 − 3x)

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Textbook Question

Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − 3y = 10, y(0) = 2

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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.

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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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