Question

Suppose that the radius of convergence of the power series $\Sigma c_{n} x^{n}$ is $R .$ What is the radius of convergence of the power series $\Sigma c_{n} x^{2 k} ?$

Name: Problem 46, Chapter 11: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2022-12-23T04:38:39-08:00
Duration: 3 min 33 s
Channel: Vincenzo Zaccaro
Description: Suppose that the radius of convergence of the power series Σcn x^n is R . What is the radius of convergence of the power series Σcn x^2 k ?

    Suppose that the radius of convergence of the power series $\Sigma c_{n} x^{n}$ is $R .$ What is the radius of convergence of the power series $\Sigma c_{n} x^{2 k} ?$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 11, Problem 46

Instant Answer

Solved by Expert Vincenzo Zaccaro

12/23/2022

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This means that the series converges for all $x$ such that $-R < x < R$. Show more…

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Suppose that the radius of convergence of the power series $\Sigma c_{n} x^{n}$ is $R .$ What is the radius of convergence of the power series $\Sigma c_{n} x^{2 k} ?$

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Key Concepts

Radius of Convergence

This concept refers to the distance from the center of a power series within which the series converges absolutely. It is determined through tests such as the ratio or root tests, and it provides a fundamental measure of the series’ behavior. Understanding the radius of convergence is essential for gauging the validity of a power series representation of a function.

Variable Substitution in Power Series

When a power series undergoes a transformation by substituting one variable for another (for example, replacing x with another function of x), the convergence properties are altered accordingly. In the case of substituting x² for x, the convergence condition changes because the exponents in the series are modified. The radius of convergence for the new series is determined by how the substitution affects the relationship between the original variable and the transformed variable.

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