Q 23 Find the indicated Maclaurin or ... [FREE SOLUTION]

Chapter 8: Q 23 (page 704)

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sqrt{x}, x_{0} = 1$

Short Answer

Expert verified

The radius of convergence for the series is $P_{n} (x) = 1 + \frac{1}{2} (x - 1) + {鈭�}_{k = 2}^{鈭�} (- 1)^{k + 1} \frac{1 路 3 路 5 鈰� (2 k - 3)}{2^{k} k!} (x - 1)^{k}$

Step by step solution

Given information

The function is $f (x) = \sqrt{x}$

Find the general of the Taylor series of the function

The Taylor series at $x = 1$ for any function $f$ with a derivative of order $n$ is given by

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

As a result, first determine the function's value as well as $f^{'} (x), f^{''} (x), f^{'''} (x)$ at $x = 1$

Furthermore, the function's general Taylor series is $P_{n} (x) = {鈭�}_{k = 0}^{鈭�} \frac{f^{k} (x_{0})}{k!} {(x - x_{0})}^{n}$

Make a table of the Taylor series for the function f(x)=x at x=1

$0$	$\sqrt{x}$	$1$	$1$
$1$	$\frac{1}{2 \sqrt{x}}$	$\frac{1}{2}$	$\frac{1}{2}$
$2$	$- \frac{1}{2^{2}} (x)^{- \frac{3}{2}}$	$- \frac{1}{2^{2}}$	$\frac{- \frac{1}{2^{2}}}{2!}$
$3$	$\frac{1 路 3}{2^{3}} (x)^{\frac{5}{2}}$	$\frac{1 路 3}{2^{3}}$	$\frac{1 路 3}{2^{3}}$
$4$	$- \frac{1 路 3 路 5}{2^{4}} (x)^{- \frac{7}{2}}$	$- \frac{1 路 3 路 5}{2^{4}}$	$- \frac{1 路 3 路 5}{2^{4}}$
.	$.$	$.$	$.$
$k$	$(- 1)^{k + 1} \frac{1 路 3 路 5 鈰� (2 k - 3)}{2^{4}} (1 - x)^{- (\frac{2 k - 1}{2})}$	$(- 1)^{k + 1} \frac{1 路 3 路 5 鈰� (2 k - 3)}{2^{k}}$	$(- 1)^{k + 1} \frac{1 路 3 路 5 鈰� (2 k - 3)}{2^{k}}$

Find the Taylor series for the function f(x)=x at x=1

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

$1 + \frac{1}{2} 路 (x - 1) + \frac{- \frac{1}{2^{2}}}{2!} {(x - 1)}^{2} + \frac{1 路 3}{2}$ ³3!(x-1)3+-1路3路52⁴4!(x-1)4+.....+(-1)k+11路3路5.....(2k-3)2^kk!(x-1)k

Or,

$P_{n} (x) = 1 + \frac{1}{2} (x - 1) + {鈭�}_{k = 2}^{鈭�} (- 1)^{k + 1} \frac{1 路 3 路 5 鈰� (2 k - 3)}{2^{k} k!} (x - 1)^{k}$

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

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sqrt{x}, x_{0} = 1$

Short Answer

Step by step solution

Given information

Find the general of the Taylor series of the function

Make a table of the Taylor series for the function f(x)=x at x=1

Find the Taylor series for the function f(x)=x at x=1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

91影视

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.x,x0=1

Short Answer

Step by step solution

Given information

Find the general of the Taylor series of the function

Make a table of the Taylor series for the function f(x)=x at x=1

Find the Taylor series for the function f(x)=x at x=1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.
$\sqrt{x}, x_{0} = 1$