/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 What is the interval of converge... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 7

What is the interval of convergence of the power series \(\Sigma(4 x)^{k} ?\)

Short Answer

Expert verified
Answer: The interval of convergence for the power series \(\Sigma(4 x)^{k}\) is \((-\infty, \frac{1}{4})\).

Step by step solution

01

Identify the general term of the series

The given power series is \(\Sigma(4 x)^{k}\). The general term of the series is \(a_k = (4x)^k\).
02

Apply the Ratio Test

We apply the Ratio Test by finding the limit of \(\frac{a_{k+1}}{a_k}\) as \(k\) approaches infinity: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \lim_{k \to \infty} \frac{(4x)^{k+1}}{(4x)^k}$$
03

Simplify the expression

Simplify the expression by dividing the terms: $$\lim_{k \to \infty} \frac{(4x)^{k+1}}{(4x)^k} = \lim_{k \to \infty} \frac{4x}{1} = 4x$$
04

Determine the condition for convergence

For the series to converge, the value of the limit must be less than 1: $$4x < 1$$ Now, solve the inequality for \(x\): $$x < \frac{1}{4}$$
05

Find the interval of convergence

The interval of convergence for the series is the set of all \(x\) values that satisfy the inequality found in Step 4: $$(-\infty, \frac{1}{4})$$ Thus, the interval of convergence for the power series \(\Sigma(4 x)^{k}\) is \((-\infty, \frac{1}{4})\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty} e^{-k x}$$

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k}$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-3 y(t)=10, y(0)=2$$

Consider the following function and its power series: $$ f(x)=\frac{1}{(1-x)^{2}}=\sum_{k=1}^{\infty} k x^{k-1}, \quad \text { for }-1

Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{x^{k}}{k(k-1)}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.