Chapter 8: Q. 53 (page 680)

In Exercises 49–56 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 41–48.
$53 . \ln (x), 3$

Short Answer

Expert verified

The Taylor series for the function $f (x) = \ln (x)$ at $x = 3$ is $P_{n} (x) = \ln 3 + \frac{1}{3} (x - 3) + \sum_{k = 2}^{\infty} (- 1)^{k + 1} \frac{1 \cdot 2 \cdot 3 \dots (k - 1)}{3^{k} k!} (x - 3)^{k}$

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,

$\begin{aligned} P_{n} (x) = & f (3) + f^{'} (3) (x - 3) + \frac{f^{''} (3)}{2!} (x - 3)^{2} + \frac{f^{'''} (3)}{3!} (x - 3)^{3} + \frac{f^{'''} (3)}{4!} (x - 3)^{4} + \dots \end{aligned}$

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

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

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	$\frac{1 \cdot 2}{x^{3}}$	$\frac{1 \cdot 2}{3^{3}}$	$\frac{\frac{1 \cdot 2}{3^{3}}}{3!}$
4	$- \frac{1 \cdot 2 \cdot 3}{x^{4}}$	$- \frac{1 \cdot 2 \cdot 3}{3^{4}}$	$\frac{- \frac{1 \cdot 2 \cdot 3}{3^{4}}}{4!}$
. . .	. . .	. . .	. . .
k	$(- 1)^{k + 1} \frac{1 \cdot 2 \cdot 3 \dots (k - 1)}{x^{k}}$	$(- 1)^{k + 1} \frac{1 \cdot 2 \cdot 3 \dots (2 k - 3)}{3^{k}}$	$(- 1)^{k + 1} \frac{\frac{1 \cdot 2 \cdot 3 \dots (k - 1)}{3^{k}}}{k!}$

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

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

Or we can write this as $P_{n} (x) = \ln 3 + \frac{1}{3} (x - 3) + \sum_{k = 2}^{\infty} (- 1)^{k + 1} \frac{1 \cdot 2 \cdot 3 \dots (k - 1)}{3^{k} k!} (x - 3)^{k}$

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In Exercises 49–56 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 41–48.
$53 . \ln (x), 3$

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

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In Exercises 49–56 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 41–48. 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

Decision Maths

Logic and Functions

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.

In Exercises 49–56 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 41–48.
$53 . \ln (x), 3$