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 $f (x) = {(\sum_{k = 0}^{\infty} \frac{x^{k + 1}}{1 + x^{2} + x})}^{2}$

Step by step solution

Given information

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

Find the Maclaurin series for the function

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

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

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

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

Find the power series for the function

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

$\frac{x^{2}}{{(1 - x^{3})}^{2}} = {(\frac{x}{1 + x^{2} + x})}^{2} {(\sum_{k = 0}^{\infty} x^{k})}^{2}$

That is

$f (x) = {(\sum_{k = 0}^{\infty} \frac{x^{k + 1}}{1 + x^{2} + x})}^{2}$

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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

Step by step solution

Given information

Find the Maclaurin series for the function

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

Find the power series for the function

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Use the Maclaurin series for 11-xto find a power series representation for x2(1-x3)2

Short Answer

Step by step solution

Given information

Find the Maclaurin series for the function

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Find the power series for the function

One App. One Place for Learning.

Most popular questions from this chapter

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