Chapter 8: Q 19 (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.
$\frac{1}{1 - x}, x_{0} = 0$

Short Answer

Expert verified

The Maclurin series for the function is $f (x) = \sum_{k = 0}^{\infty} x^{k}$

Step by step solution

Given information

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

Find the general of the Maclurin series of the function

The Maclurin series at $x_{0} = 0$ for any function $f$ with a derivative of all orders is given by

$f (x) = f (0) + f^{'} (0) x + \frac{f^{''} (0)}{2!} x^{2} + \frac{f^{'''} (0)}{3!} x^{3} + \frac{f^{''''} (0)}{4!} x^{4} + \dots$

The function's generic form Maclurin series can be written.

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

Make a table of the Maclurin series for the function f(x)=11-x at x=0

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

Find the Maclaurin series for the function f(x)=11-x at x=0

The Maclaurin series for the function $f (x) = \frac{1}{1 - x}$ at $x = 0$ is:

$1 + x + x^{2} + x^{3} + \dots + x^{k} + \dots$

Or we can write as:

$f (x) = \sum_{k = 0}^{\infty} x^{k}$

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

Short Answer

Step by step solution

Given information

Find the general of the Maclurin series of the function

Make a table of the Maclurin series for the function f(x)=11-x at x=0

Find the Maclaurin series for the function f(x)=11-x at x=0

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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.11-x,x0=0

Short Answer

Step by step solution

Given information

Find the general of the Maclurin series of the function

Make a table of the Maclurin series for the function f(x)=11-x at x=0

Find the Maclaurin series for the function f(x)=11-x at x=0

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Calculus

Decision Maths

Statistics

Logic and Functions

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