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 8: Q. 23 (page 700)

Show that when you take the derivative of the Maclaurin series for the exponential function term by term, you obtain the same series you started with. Why does that make sense?

Short Answer

Expert verified

The Maclaurin series derivate for the exponential function term by term is the same series as before.

Step by step solution

01

Step 1. Given information

The function is $f (x) = e^{x}$

02

Step 2. Calculation

Let us consider the function $f (x) = e^{x}$

For function, the Maclaurin series is as follows:

$e^{x} = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{2 k}$

The derivative of the function $f (x)$ is

$\begin{array}{rcl} f^{'} (x) & = & \frac{d}{d x} (\sum_{k = 0}^{\infty} \frac{1}{k!} x^{k}) \\ = & \sum_{k = 0}^{\infty} \frac{1}{k!} \frac{d}{d x} (x)^{k} \\ = & \sum_{k = 0}^{\infty} \frac{1}{k!} k \cdot x^{k - 1} \\ = & \sum_{k = 0}^{\infty} \frac{1}{(k - 1)!} x^{k - 1} \end{array}$

It can also be written as:

$f^{'} (x) = 0 + 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!}$

Therefore, the Maclaurin series derivate for the exponential function term by term is the same series as before.

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

In Exercises 41–48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$

$46 . \sqrt[3]{x}, 1$

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:

$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Find and graph the first four terms in the sequence of partial sums of $J_{o} (x)$ .

Show that $\sum_{k = 0}^{\infty} \frac{1}{1 + 2^{k}} x^{k}$ , the power series in $x$ from Example 1, diverges when $x = - 2$

Why is it helpful to know the Maclaurin series for a few basic functions?

What is $x_{0}$ if the interval of convergence for the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k} is (2, 10] ?$

See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

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