Q. 53 In Exercises 49鈥�56 find the Ta... [FREE SOLUTION]

Chapter 8: Q. 53 (page 680)

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41鈥�48.
$53 . \ln (x), 3$

Short Answer

Expert verified

The Taylor series for the function $f (x) = \ln (x)$ at $x = 3$ is $P_{n} (x) = \ln 3 + \frac{1}{3} (x 鈭� 3) + {鈭�}_{k = 2}^{鈭�} (鈭� 1)^{k + 1} \frac{1 鈰� 2 鈰� 3 鈰� (k 鈭� 1)}{3^{k} k!} (x 鈭� 3)^{k}$

Step by step solution

Step 1. Given data

We have the function $f (x) = \ln (x)$

Step 2. Table of the taylor series

Any function $f$ with a derivative of order $n$ , the taylor series at $x = 3$ is given by,

$\begin{aligned} P_{n} (x) = & f (3) + f^{鈥�} (3) (x 鈭� 3) + \frac{f^{鈥测赌�} (3)}{2!} (x 鈭� 3)^{2} + \frac{f^{鈥测赌� 鈥�} (3)}{3!} (x 鈭� 3)^{3} + \frac{f^{鈥测赌� 鈥�} (3)}{4!} (x 鈭� 3)^{4} + 鈥� \end{aligned}$

we can write the general of the Taylor series of the function $f$ is,

$P_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{f^{k} (x_{0})}{k!} {(x 鈭� x_{0})}^{n}$

So, let us first construct the table of the Taylor series for the function $f (x) = \ln (x)$ at $x = 3$ .

n	$f^{n} (x)$	$f^{n} (3)$	$\frac{f^{n} (3)}{n!}$
0	$\ln (x)$	$\ln (3)$	$\ln (3)$
1	$\frac{1}{x}$	$\frac{1}{3}$	$\frac{1}{3}$
2	$- \frac{1}{x^{2}}$	$- \frac{1}{3^{2}}$	$\frac{- \frac{1}{3^{2}}}{2!}$
3	$\frac{1 鈰� 2}{x^{3}}$	$\frac{1 鈰� 2}{3^{3}}$	$\frac{\frac{1 鈰� 2}{3^{3}}}{3!}$
4	$鈭� \frac{1 鈰� 2 鈰� 3}{x^{4}}$	$鈭� \frac{1 鈰� 2 鈰� 3}{3^{4}}$	$\frac{鈭� \frac{1 鈰� 2 鈰� 3}{3^{4}}}{4!}$
. . .	. . .	. . .	. . .
k	$(鈭� 1)^{k + 1} \frac{1 鈰� 2 鈰� 3 鈰� (k 鈭� 1)}{x^{k}}$	$(鈭� 1)^{k + 1} \frac{1 鈰� 2 鈰� 3 鈰� (2 k 鈭� 3)}{3^{k}}$	$(鈭� 1)^{k + 1} \frac{\frac{1 鈰� 2 鈰� 3 鈰� (k 鈭� 1)}{3^{k}}}{k!}$

Step 3. Taylor series for the f(x)=lnx

The Taylor series for the function $f (x) = \ln (x)$ at $x = 3$ is $\begin{aligned} P_{n} (x) = & \ln 3 + \frac{1}{3} 鈰� (x 鈭� 3) + \frac{鈭� \frac{1}{3^{2}}}{2!} (x 鈭� 3)^{2} + \frac{\frac{1 鈰� 2}{3^{3}}}{3!} (x 鈭� 3)^{3} + \frac{鈭� \frac{1 鈰� 2 鈰� 3}{3^{4}}}{4!} (x 鈭� 3)^{4} + 鈥� + \frac{(鈭� 1)^{k + 1} \frac{1 鈰� 2 鈰� 3 鈰� (k 鈭� 1)}{3^{k}}}{k!} (x 鈭� 3)^{k} \end{aligned}$

Or we can write this as $P_{n} (x) = \ln 3 + \frac{1}{3} (x 鈭� 3) + {鈭�}_{k = 2}^{鈭�} (鈭� 1)^{k + 1} \frac{1 鈰� 2 鈰� 3 鈰� (k 鈭� 1)}{3^{k} k!} (x 鈭� 3)^{k}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41鈥�48.
$53 . \ln (x), 3$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Table of the taylor series

Step 3. Taylor series for the f(x)=lnx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48. 53.lnx,3

Short Answer

Step by step solution

Step 1. Given data

Step 2. Table of the taylor series

Step 3. Taylor series for the f(x)=lnx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41鈥�48.
$53 . \ln (x), 3$