Chapter 8: Q 21 (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.
$\frac{1}{x}, x_{0} = 1$

Short Answer

Expert verified

The radius of convergence for the series is $P_{n} (x) = \sum_{k = 0}^{\infty} (- 1)^{k} (x - 1)^{k}$

Step by step solution

Given information

The function is $f (x) = \frac{1}{x}$

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

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

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

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

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

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

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

$1 - 1 (x - 1) + 1 (x - 1)^{2} + \dots + (- 1)^{k} (x - 1)^{k} + \dots$

Or, $P_{n} (x) = \sum_{k = 0}^{\infty} (- 1)^{k} (x - 1)^{k}$

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

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)=1x at x=1

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

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Discrete Mathematics

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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.
$\frac{1}{x}, x_{0} = 1$