Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks

Exams
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

EXAM TYPES

GCSE

IGCSE

AS

A Level

International A Level

University Admissions Tests

GCSE SUBJECTS

GCSE 1

GCSE 2

GCSE 3

GCSE 4

GCSE 5

SOME-TEXT

IGCSE SUBJECTS

IGCSE 1

AS SUBJECTS

AS 1

A Level SUBJECTS

A Level 1

International A Level SUBJECTS

International A Level 1

University Admissions Tests SUBJECTS

University Admissions Tests 1
Features
Features

Discover all of these amazing features with a free account.

Flashcards

Vaia AI

Notes

Study Plans

Study Sets
What’s 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.
Resources
Discover

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

Find a job

Find a degree

Student Deals

Magazine

Mobile App
Featured

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

Job Board
The largest student job board with the most exciting opportunities.

Vaia Deals
Verified student deals from top brands.

Our App
Discover our mobile app to take your studies anywhere.

Go to App

Learning Materials

Features

Discover

Chapter 1: Q6P (page 32)

$x \sqrt{1 + x}$

Short Answer

Expert verified

The series is $x \sqrt{1 + x} = \sum_{n = 2}^{\infty} \frac{(- 1)^{n - 1} (2 n - 3)!!}{(2 n)!!} x^{n + 1}$

The sum is given below.

$S_{1} = x$

$S_{2} = x + \frac{1}{2} x^{2}$

$S_{3} = x + \frac{1}{2} x^{2} - \frac{1}{8} x^{3}$

$S_{4} = x + \frac{1}{2} x^{2} - \frac{1}{8} x^{3} + \frac{1}{16} x^{4}$

The graphs are shown below.

Step by step solution

01

Given Information

The Maclaurin series.

02

Definition of the MaclaurinSeries.

A Maclaurin series is a function with an expansion series that gives the sum of the function's derivatives.

03

Find the series and sum.

The Maclaurin series is given below.

$(1 + x)^{1 / 2} = 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{16} x^{3} - \frac{5}{128} x^{4} + \dots$

$x (1 + x)^{1 / 2} = x + \frac{1}{2} x^{2} - \frac{1}{8} x^{3} + \frac{1}{16} x^{4} - \frac{5}{128} x^{5} + \dots$

The general series is given below

$x \sqrt{1 + x} = x \sum_{n = 0}^{\infty} (\begin{matrix} \frac{1}{2} \\ n \end{matrix}) x^{n}$

$x \sqrt{1 + x} = \sum_{n = 2}^{\infty} \frac{(- 1)^{n - 1} (2 n - 3)!!}{(2 n)!!} x^{n + 1}$

The sum is given below.

localid="1657352249901" $S_{1} = x$

$S_{2} = x + \frac{1}{2} x^{2}$

$S_{3} = x + \frac{1}{2} x^{2} - \frac{1}{8} x^{3}$

$S_{4} = x + \frac{1}{2} x^{2} - \frac{1}{8} x^{3} + \frac{1}{16} x^{4}$

04

Plot the graphs.

The graph of the function with $S_{1}, S_{2}, S_{3}, S_{4}$ is given below

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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.

$\lim_{x \to 0} \frac{In (1 - x)}{x}$ .

Consider the series in Problem 4.6and show that the remainder after n terms is $R_{n} = \frac{1}{n + 1}$ . Compare the value of term $n + 1$ with $R_{n}$ for n=3, n=10, n=100, n=500to see that the first neglected term is not a useful estimate of the Error.

Prove that an absolutely convergent series $\sum_{m = 1}^{\infty} a_{n}$ is convergent. Hint: Put $b_{n} = a_{n} + | a_{n} |$ . Then theare nonnegative; we have $| b_{n} | \leq 2 | a_{n} |$ and $a_{n} = b_{n} - | a_{n} |$

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6. $\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$

$\arcsin x = \int_{0}^{x} \frac{d t}{\sqrt{1 - t^{2}}}$

See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Thermodynamics

Read Explanation

Rotational Dynamics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Famous Physicists

Read Explanation

Further Mechanics and Thermal 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.

Sign-up for free