Chapter 8: Q. 32 (page 680)

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

Short Answer

Expert verified

The Maclaurin series is,

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

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

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,

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

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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.

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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

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