/*! 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 74. Let f聽be an even function with ... [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 8: Q 74. (page 702)

Let $f$ be an even function with Maclaurin series representation ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$ . Prove thatrole="math" localid="1650365708378" $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .

Short Answer

Expert verified

The solution is $a_{2 k + 1} = 0$ for every nonnegative integer $k$ .

Step by step solution

01

Step 1. Given information.

The given even function is:

$f (x) = {鈭�}_{k = 0}^{鈭�} a_{k} x^{k}$

02

Step 2. Prove the given statement.

It is given that $f (x)$ is an even function then $f (x) = f (- x)$ .

Now evaluate $f (- x)$ .

role="math" localid="1650366151821" $\begin{array}{rcl} f (- x) & = & {鈭�}_{k = 0}^{鈭�} a_{k} {(- x)}^{k} \\ = & {鈭�}_{k = 0}^{鈭�} (- 1)^{k} a_{k} x^{k} \end{array}$

Since $f (x) = f (- x)$ , implies that ${鈭�}_{k = 0}^{鈭�} a_{k} x^{k} \begin{array}{rcl} = \end{array} \begin{array}{rcl} {鈭�}_{k = 0}^{鈭�} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} ( \end{array} \begin{array}{rcl} - \end{array} \begin{array}{rcl} 1 \end{array} \begin{array}{rcl} )^{k} \end{array} \begin{array}{rcl} a_{k} \end{array} \begin{array}{rcl} x^{k} \end{array}$ .

That is, $a_{k} = {(- 1)}^{k} a_{k}$

We know that it is possible only when $a_{2 k + 1} = 0$ , for every value of $k$ .

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

Exercise 64-68 concern with the bessel function.

What is the interval for convergence for $J_{0} (x) ?$

Show that the power series ${鈭�}_{k = 0}^{鈭�} 鈥� \frac{(鈭� 1)^{k}}{2 k + 1} x^{2 k + 1}$ converges conditionally when $x = 1$ and when $x = - 1$ . What does this behavior tell you about the interval of convergence for the series?

Find the interval of convergence for power series: ${鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} x^{2 k + 1}$

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

${鈭�}_{k = 0}^{鈭�} {(- 1)}^{k} \frac{k^{2}}{3^{k}} {(x - \frac{1}{2})}^{k}$

How may we find the Maclaurin series for f(x)g(x) if we already know the Maclaurin series for the functions f(x) and g(x)? How do you find the interval of convergence for the new series?

See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.