Q. 32 Find the Maclaurin series for th... [FREE SOLUTION]

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Chapter 8: Q. 32 (page 680)

Find the Maclaurin series for the specified function:
$e^{x}$ .

Short Answer

Expert verified

The Maclaurin series is,

$f (x) = {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} x^{k}$ .

Step by step solution

Step 1. Given Information.

The function is $e^{x}$ .

Step 2. Finding the Maclaurin series.

Let $f (x) = e^{x}$ .

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

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

Or we can write the general form Maclaurin series of the function $f$ is, $f (x) = {鈭�}_{n = 0}^{鈭�} \frac{f^{n} (0)}{n!} x^{n}$ .

The table of the Maclaurin series for the function $f (x) = e^{x}$

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

The Maclaurin series is,

$f (x) = {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} x^{k}$ .

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91影视

Find the Maclaurin series for the specified function:
$e^{x}$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the Maclaurin series.

One App. One Place for Learning.

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91影视

Find the Maclaurin series for the specified function:ex.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the specified function:
$e^{x}$ .