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Chapter 8: Q 30 (page 704)

Use the Maclaurin series for $\frac{1}{1 - x}$ to find a power series representation for $\frac{x^{2}}{{(1 - x^{3})}^{2}}$

Short Answer

Expert verified

The power series for the function is $f (x) = {({鈭�}_{k = 0}^{鈭�} \frac{x^{k + 1}}{1 + x^{2} + x})}^{2}$

Step by step solution

01

Given information

$f (x) = \frac{x^{2}}{{(1 - x^{3})}^{2}}$

02

Find the Maclaurin series for the function

$g (x) = \frac{1}{1 - x}$ at $x = 0$ is

$1 + x + x^{2} + x^{3} + 鈰� + x^{k} + 鈰�$

Or $f (x) = {鈭�}_{k = 0}^{鈭�} x^{k}$

03

The function f(x)=x21-x32 is written as:

$\frac{x^{2}}{{(1 - x^{3})}^{2}} = \frac{x^{2}}{(1 - x^{2}) {(1 + x^{2} + x)}^{2}} = {(\frac{x}{1 + x^{2} + x})}^{2} \frac{1}{(1 - x)}$ ²

04

Find the power series for the function

The power series for the function $f (x) = \frac{x^{2}}{{(1 - x^{3})}^{2}}$ is

$\frac{x^{2}}{{(1 - x^{3})}^{2}} = {(\frac{x}{1 + x^{2} + x})}^{2} {({鈭�}_{k = 0}^{鈭�} x^{k})}^{2}$

That is

$f (x) = {({鈭�}_{k = 0}^{鈭�} \frac{x^{k + 1}}{1 + x^{2} + x})}^{2}$

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Most popular questions from this chapter

Given a function f and a Taylor polynomial for fat $x_{0}$ , what is meant by the nth remainder $R_{n} (x)$ ? What does $R_{n} (x)$ measure?

Exercise 64-68 concern with the bessel function.

What is the interval for convergence for $J_{0} (x) ?$

If m is a positive integer, how can we find the Maclaurin series for the function $f (c x^{m})$ if we already know the Maclaurin series for the function f(x)? How do you find the interval of convergence for the new series?

If f(x) is an nth-degree polynomial and $P_{n} (x)$ is the nth Taylor polynomial for fat $x_{0}$ , what is the nth remainder $R_{n} (x)$ ? What is $R_{n + 1} (x)$ ?

Find the interval of convergence for power series: ${鈭�}_{k = 1}^{鈭�} {(- 1)}^{K} \frac{1}{k^{k}} {(x - 1)}^{k}$

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