Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.2.2

Capitolo 11, Problema 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.

Guida verificata passo dopo passo

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 \(\sum_{k=0}^\infty c_k (x - 4)^k\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

05:58

Intro to 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.

Video consigliato:

05:58

Intro to Power 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.

Video consigliato:

05:58

Intro to Power Series

Pratica correlata

Domanda del libro di testo

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

views

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) = ln √(1 − x²)

views

Domanda del libro di testo

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)

views

Domanda del libro di testo

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

views

Domanda del libro di testo

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.

views

Domanda del libro di testo

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

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

views