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Chapter 11: Problem 6

Find a power series in \(x\) that has the given sum, and specify the radius of convergence. (Hint: Use \((11.15),(11.40),\) or long division, as necessary.) $$\frac{x}{1-x^{4}}$$

Short Answer

Expert verified
The power series is \( x + x^5 + x^9 + x^{13} + \ldots \) with a radius of convergence of 1.

Step by step solution

01

Understand the Given Function

The problem asks us to find a power series for the function \(\frac{x}{1-x^{4}}\). This function is a modification of the standard geometric series formula \(\frac{1}{1-a}\) where \(a=x^{4}\).
02

Recall the Geometric Series Formula

The geometric series is given by \( \frac{1}{1-a} = 1 + a + a^2 + a^3 + \ldots \) for \(|a| < 1\). Here, \(a = x^4\).
03

Apply the Geometric Series to the Denominator

We write \( \frac{1}{1-x^4} = 1 + x^4 + (x^4)^2 + (x^4)^3 + \ldots \) which simplifies to \(1 + x^4 + x^8 + x^{12} + \ldots\).
04

Multiply by the Numerator \(x\)

Next, multiply the series by \(x\) to fit the original function: \[ x \cdot \left( 1 + x^4 + x^8 + x^{12} + \ldots \right) = x + x^5 + x^9 + x^{13} + \ldots \].
05

Determine the Radius of Convergence

The original geometric series \( \frac{1}{1-x^4} \) converges for \( |x^4| < 1 \), i.e., \( |x| < 1 \). Thus, the radius of convergence for \( \frac{x}{1-x^4} \) is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is an essential mathematical concept that arises frequently in different contexts. It is a sequence of numbers where each term is found by multiplying the previous term by a constant, called the common ratio. A simple formula for the sum of an infinite geometric series is \( \frac{1}{1-a} = 1 + a + a^2 + a^3 + \ldots \) for \(|a| < 1\).
  • The geometric series formula can be very useful when working with a function of the form \(\frac{1}{1-a}\).
  • The condition \(|a| < 1\) ensures that the series converges to a finite sum.
  • For the function we are discussing, the term \(a\) is replaced with \(x^4\), leading to a modified series.
This concept is crucial for transforming functions into power series, allowing us to express complicated functions in simpler forms.
Radius of Convergence
When dealing with power series, determining where the series converges is crucial. The radius of convergence defines the interval over which a power series converges. It is found by considering \(|x-a| < R\), where \(R\) is the radius and \(a\) is the center of the series.
  • In simpler terms, the radius of convergence tells us how far from the center point the series still converges.
  • Based on our function, \(|x^4| < 1\) simplifies to \(|x| < 1\), hence the radius of convergence is 1.
  • This means that the power series will converge for any \(x\) within the interval from -1 to 1.
Understanding the radius of convergence helps ensure we are using the series in its valid range.
Series Convergence
Convergence of a series means reaching a sum, or a specific value, as you add more terms. For a function represented by a power series, convergence is vital for the series to accurately model the function.
  • The geometric series provides a basis for understanding how different types of series converge.
  • Convergence ensures that even though we are adding infinite terms, they sum to a finite number.
  • For example, the convergence of the series for \(\frac{x}{1-x^4}\) allows us to express it reliably as \(x + x^5 + x^9 + x^{13} + \ldots\) for \(|x| < 1\).
Ensuring that a series converges within a certain range allows us to make valid predictions and calculations based on the series.
Mathematical Function Expansion
Many functions that might initially seem complex can be simplified through mathematical function expansion. This technique involves representing a function as a series.
  • Power series are the most commonly used method for function expansion.
  • This allows for easier manipulation and straightforward computation of function values.
  • In the given example, \(\frac{x}{1-x^4}\), using series expansion simplifies the function to \(x + x^5 + x^9 + x^{13} + \ldots\).
Utilizing function expansion can make complex mathematical tasks more manageable and can aid in solving various problems efficiently.

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Most popular questions from this chapter

Determine whether the series converges or diverges. \(\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{n}}\)

Show that \(\cos x \approx 1-\frac{1}{2} x^{2}+\frac{1}{24} x^{4}-\frac{1}{720} x^{6}\) is accurate to five decimal places if \(0 \leq x \leq \pi / 4\).

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If the series is positive-term, determine whether it is convergent or divergent; if the series contains negative terms, determine whether it is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{(0.9)^{n}}{\ln n} $$

(a) Let \(g(x)\) be the sum of the first two nonzero terms of the Maclaurin series for \(f(x) .\) Use \(g(x)\) to approximate \(\int_{0}^{1} f(x) d x\) and \(\int_{1}^{2} f(x) d x .\) (b) Graph, on the same coordinate axes, \(f\) and \(g\) for \(0 \leq x \leq 2,\) and then use the graphs to compare the accuracy of the approximations in (a). \(f(x)=\sin \left(x^{2}\right)\)
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