Q. 34 ind the Maclaurin series for e-x... [FREE SOLUTION]

91影视

Chapter 8: Q. 34 (page 704)

ind the Maclaurin series for $e^{- x^{2}}$ , and use it to approximate ${鈭�}_{0}^{1} e^{- x^{2}} d x$ to within 0.001 of its value. How many terms would you need to approximate the integral to within $10^{- 6}$ of its value?

Short Answer

Expert verified

The Maclaurin series can be given as

$e^{- x^{2}} = 1 - x^{2} + \frac{x^{4}}{2} - \frac{x^{6}}{3!} + \frac{x^{8}}{4!} - . . .$

On integrating we get,

${鈭�}_{0}^{1} e^{- x^{2}} d x = 1 - \frac{1}{3} + \frac{1}{10} - \frac{1}{42} + \frac{1}{216} - . . .$

And we require first 6 terms to get corrected values upto 0.0001

Step by step solution

Given information

We are given the power series of $e^{x}$

Find the power series of e-x2

We know the Maclaurin series of $e^{x}$ which can be given as

role="math" localid="1654420433127" $e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + . . . s u b s t i t u t e x = - x^{2} e^{- x^{2}} = 1 - x^{2} + \frac{x^{4}}{2} - \frac{x^{6}}{3!} + \frac{x^{8}}{4!} - . . . N o w i n t e g r a t i n g {鈭�}_{0}^{1} e^{- x^{2}} d x = {鈭�}_{0}^{1} 1 - x^{2} + \frac{x^{4}}{2} - \frac{x^{6}}{3!} + \frac{x^{8}}{4!} - . . . d x {鈭�}_{0}^{1} e^{- x^{2}} d x = {[x - \frac{x^{3}}{3} + \frac{x^{5}}{10} - \frac{x^{7}}{42} + \frac{x^{9}}{216} - . . . . . .]}^{1}_{0} {鈭�}_{0}^{1} e^{- x^{2}} d x = 1 - \frac{1}{3} + \frac{1}{10} - \frac{1}{42} + \frac{1}{216} - . . .$

We reuqire first 6 terms to get values corrected upto 0.0001 of its value

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

ind the Maclaurin series for $e^{- x^{2}}$ , and use it to approximate ${鈭�}_{0}^{1} e^{- x^{2}} d x$ to within 0.001 of its value. How many terms would you need to approximate the integral to within $10^{- 6}$ of its value?

Short Answer

Step by step solution

Given information

Find the power series of e-x2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Geometry

Applied Mathematics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

91影视

ind the Maclaurin series for e-x2, and use it to approximate 鈭�01e-x2dx to within 0.001 of its value. How many terms would you need to approximate the integral to within10-6 of its value?

Short Answer

Step by step solution

Given information

Find the power series of e-x2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Geometry

Applied Mathematics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

ind the Maclaurin series for $e^{- x^{2}}$ , and use it to approximate ${鈭�}_{0}^{1} e^{- x^{2}} d x$ to within 0.001 of its value. How many terms would you need to approximate the integral to within $10^{- 6}$ of its value?