Chapter 8: Q. 31 (page 680)

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

Short Answer

Expert verified

The Maclaurin series is,

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

Step by step solution

Step 1. Given Information.

The function is,

$\cos (x)$ .

Step 2. Formulae the Maclaurin series.

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

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

localid="1649791589053" $f (x) = f (0) + f' (0) x + \frac{f'' (0)}{2!} x^{2} + \frac{f''' (0)}{3!} x^{3} + \frac{f'''' (0)}{4!} x^{4} + . . . . .$

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

$f (x) = \sum_{n = 0}^{\infty} \frac{f^{n} (0)}{n!} x^{n}$

Step 3. Finding the Maclaurin series.

The table of the Maclaurin series for the function is,

The Maclaurin series for the function $f (x) = \cos (x)$ is,

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

Or, we can write it as,

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

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulae the Maclaurin series.

Step 3. Finding the Maclaurin series.

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Find the Maclaurin series for the specified function:cosx.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formulae 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

Calculus

Logic and Functions

Mechanics Maths

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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