Chapter 8: Q.33 (page 680)

Find the Maclaurin series for the specified function:
$\sin (x)$ .

Short Answer

Expert verified

The Maclaurin series is,

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

Step by step solution

Step 1. Given Information.

The function is,

$\sin (x)$ .

Step 2. Formulae for the Maclaurin series.

Let $f (x) = \sin (x)$ .

Since, for any function $f$ with derivatives of all orders at the point $x_{0} = 0$ then the Maclaurin series is,

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

Or, we can write the general form Maclaurin series of the function $f$ is,

localid="1649653268867">f(x)=∑n=0∞fn(0)n!xn.

Step 3. Finding the Maclaurin series.

So, let us first construct the table of the Maclaurin series for the function $f (x) = \sin (x)$ ,

Therefore the Maclaurin series for the function $f (x) = \sin (x)$ is,

$0 + 1 . x + \frac{0}{2!} x^{2} + \frac{(- 1)}{3!} x^{3} + 0 x^{4} + \frac{1}{5!} x^{5} + \frac{0}{6!} x^{6} + . . . . . .$

Or, we can write as,

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

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the Maclaurin series for the specified function:
$\sin (x)$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulae for the Maclaurin series.

Step 3. Finding the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Probability and Statistics

Pure Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the specified function:sinx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulae for the Maclaurin series.

Step 3. Finding the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Probability and Statistics

Pure Maths

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the specified function:
$\sin (x)$ .