/*! 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} Q.10.9P 鈭憂=1鈭�(-1)nn3xn... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Chapter 1: Q.10.9P (page 22)

${鈭�}_{n = 1}^{鈭�} {(- 1)}^{n} n^{3} x^{n}$

Short Answer

Expert verified

The interval of convergence is $(- 1, 1)$

Step by step solution

01

Given Information

The power series is ${鈭�}_{n = 1}^{鈭�} {(- 1)}^{n} n^{3} x^{n}$

02

Definition of the interval of convergence.

The Interval of convergence is the interval in which the power series is convergent.

03

Find the interval.

The power series is ${鈭�}_{n = 1}^{鈭�} {(- 1)}^{n} n^{3} x^{n}$

Let $p_{n} = |\frac{a_{n - 1}}{a_{n}}|$ .

Substitute the value of the power series in the formula above, the equation becomes as follows,

$蚁_{n} = |\frac{a_{n + 1}}{a_{n}}|$

$= |\frac{(n + 1)^{3} x^{n + 1}}{n^{3} x^{n}}|$

$= |\frac{(n + 1)^{3} x}{n^{3}}|$

Apply limits in the above equation,

$蚁 \lim_{n 鈫� 鈭�} = |{(1 + \frac{1}{n})}^{3} x|$

$= |x|$

The power series is convergent for $蚁 < 1$ ,

Hence the given power series is convergent for $|x| < 1$ and divergent for $|x| > 1$ .

Now check the ends points,

When $x = 1$ series is ${鈭�}_{n = 1}^{鈭�} {(- 1)}^{n} n^{3}$ , which is a divergent alternating series.

When $x = - 1$ series is ${鈭�}_{n = 1}^{鈭�} n^{3}$ , which is a divergent series.

Hence the interval of convergence is $(- 1, 1)$ .

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

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{d^{4}}{d x^{4}} \ln (1 + x^{3}) a t x = 0$ .

$\sqrt{\frac{1 - x}{1 + x}}$

Find the values of several derivatives of $e^{- 1 / t^{2}}$ at t = 0. Hint:Calculate a few derivatives (as functions of t); then make the substitution $x = 1 / t^{2}$ , and use the result of Problem 24(f) or 25.

$\sqrt{1 + \ln (1 + x)}$

$In \cos x$ Hints:Method1:Write $\cos x = 1 + (\cos x - 1) = 1 + u$ ;use the series you know for $In (1 + u)$ ;replace u by the Maclaurin series for $(\cos x - 1)$

Method2: $In \cos x =$ $- {鈭�}_{0}^{x} \tan udu$ Use the series of Example 2 in method B.

See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Particle Model of Matter

Read Explanation

Solid State Physics

Read Explanation

Thermodynamics

Read Explanation

Waves Physics

Read Explanation

Kinematics Physics

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.