Chapter 8: Q. 54 (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.
$54 . \sqrt[3]{x}, 1$

Short Answer

Expert verified

The Taylor series for the function $f (x) = \sqrt[3]{x}$ at $x = 1$ is $P_{n} (x) = 1 + \frac{1}{3} (x - 1) + \sum_{k = 2}^{\infty} (- 1)^{k + 1} \frac{5 \cdot 8 \cdot 11 \dots (2 k - 3)}{3^{k} k!} (x - 1)^{k}$

Step by step solution

Step 1. Given data

We have the function $f (x) = \sqrt[3]{x}$

Step 2. Table of the taylor series

Any function $f$ with a derivative of order $n$ , the taylor series at $x = 1$ is given by,

$P_{n} (x) = f (1) + f^{'} (1) (x - 1) + \frac{f^{''} (1)}{2!} (x - 1)^{2} + \frac{f^{''} (1)}{3!} (x - 1)^{3} + \frac{f^{''''} (1)}{4!} (x - 1)^{4} + \dots$

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 $f (x) = \sqrt[3]{x}$ at $x = 1$

n	$f^{n} (x)$	$f^{n} (0)$	$\frac{f^{n} (0)}{n!}$
0	$\sqrt[3]{x}$	1	1
1	$\frac{1}{3} x^{\frac{- 2}{3}}$	$\frac{1}{3}$	$\frac{1}{3}$
2	$- \frac{2}{3^{2}} (x)^{- \frac{5}{3}}$	$- \frac{2}{3^{2}}$	$\frac{- \frac{2}{3^{2}}}{2!}$
3	$\frac{2 \cdot 5}{3^{3}} (x)^{- \frac{8}{3}}$	$\frac{2 \cdot 5}{3^{3}}$	$\frac{\frac{2 \cdot 5}{3^{3}}}{3!}$
4	$- \frac{2 \cdot 5 \cdot 8}{3^{4}} (x)^{- \frac{11}{3}}$	$- \frac{2 \cdot 5 \cdot 8}{3^{4}}$	$\frac{- \frac{2 \cdot 5 \cdot 8}{3^{4}}}{4!}$
. . .	. . .	. . .	. . .
k	$(- 1)^{k + 1} \frac{2 [5 \cdot 8 \dots (2 k - 3)]}{3^{k}} (x)^{- (\frac{3 k - 1}{2})}$	$(- 1)^{k + 1} \frac{2 [5 \cdot 8 \dots (2 k - 3)]}{3^{k}}$	$(- 1)^{k + 1} \frac{\frac{2 [5 \cdot 8 \dots (2 k - 3)]}{3^{k}}}{k!}$

Step 3. Taylor series for the f(x)=x3

The Taylor series for the function $f (x) = \sqrt[3]{x}$ at $x = 1$ is

$P_{n} (x) = \begin{aligned} 1 & + \frac{1}{3} \cdot (x - 1) + \frac{- \frac{2}{3^{2}}}{2!} (x - 1)^{2} + \frac{\frac{2 \cdot 5}{3^{3}}}{3!} (x - 1)^{3} + \frac{- \frac{2 \cdot 5 \cdot 8}{3^{4}}}{4!} (x - 1)^{4} + \dots + \frac{(- 1)^{k + 1} \frac{5 \cdot 8 \cdot 11 \dots (2 k - 3)}{3^{k}}}{k!} (x - 1)^{k} \end{aligned}$

Or we can write this as $P_{n} (x) = 1 + \frac{1}{3} (x - 1) + \sum_{k = 2}^{\infty} (- 1)^{k + 1} \frac{5 \cdot 8 \cdot 11 \dots (2 k - 3)}{3^{k} k!} (x - 1)^{k}$

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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.
$54 . \sqrt[3]{x}, 1$

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)=x3

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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. 54.x3,1

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)=x3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

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.
$54 . \sqrt[3]{x}, 1$