Chapter 1: Q17MP (page 45)

Find the Maclaurin series of the following functions.
$e^{- 1 \sqrt{1 - x^{2}}}$

Short Answer

Expert verified

The Maclaurin series of $e^{- 1 \sqrt{1 - x^{2}}}$ is $1 + \frac{x^{2}}{2} + \frac{x^{4}}{4} + \frac{7 x^{6}}{48} + \dots$ .

Step by step solution

Maclaurin series and the given function:

Function is $e^{- 1 \sqrt{1 - x^{2}}}$ ,

The Maclaurin series of $e^{x}$ is expressed as follows:

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots$

The Maclaurin series of is expressed as follows:

role="math" localid="1658895933390" ${(1 + x)}^{1 / 2} = 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{16} x^{3} - \frac{5}{128} x^{4} + \dots$

Calculation by the Maclaurin series:

Replace x by $- x^{2}$ in the Maclaurin series of ${(1 + x)}^{1 / 2}$ as follows:

${(1 - x^{2})}^{\frac{1}{2}} = 1 - \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{16} x^{6} + \dots {(1 - x^{2})}^{\frac{1}{2}} = \frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots$

Therefore, the Maclaurin series of $1 - {(1 - x^{2})}^{\frac{1}{2}}$ is $\frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots$

Substitute $\frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots$ for x in expansion of $e^{x}$ as follows:

role="math" localid="1658897078635" $e^{- 1 \sqrt{1 - x^{2}}} = [\begin{matrix} 1 + [\frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots] + \frac{{[\frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots]}^{2}}{2!} \\ + \frac{[\frac{1}{2} x^{2} + \frac{1}{8} x^{4} + \frac{1}{16} x^{6} + \dots]}{3!} \end{matrix}] e^{- 1 \sqrt{1 - x^{2}}} = [1 + \frac{192 x^{2} + 96 x^{4} + 56 x^{6}}{384} + . . . .] = 1 + \frac{x^{2}}{2} + \frac{x^{4}}{4} + \frac{7 x^{6}}{48} + . . . . . .$

Hence, the Maclaurin series of $e^{- 1 \sqrt{1 - x^{2}}}$ is $1 + \frac{x^{2}}{2} + \frac{x^{4}}{4} + \frac{7 x^{6}}{48} + . . . . . .$

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!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the Maclaurin series of the following functions.
$e^{- 1 \sqrt{1 - x^{2}}}$

Short Answer

Step by step solution

Maclaurin series and the given function:

Calculation by the Maclaurin series:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Work Energy and Power

Waves Physics

Thermodynamics

Turning Points in Physics

Space Physics

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series of the following functions.e-11-x2

Short Answer

Step by step solution

Maclaurin series and the given function:

Calculation by the Maclaurin series:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Work Energy and Power

Waves Physics

Thermodynamics

Turning Points in Physics

Space Physics

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series of the following functions.
$e^{- 1 \sqrt{1 - x^{2}}}$