Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series (see Examples 1 and 2). $$ f(x)=\frac{x^{3}}{2-x^{3}} $$

Step-by-Step Solution

Answer

The power series is \( \sum_{n=0}^{\infty} \frac{x^{3n}}{2^{n+1}} \) with radius \( \sqrt[3]{2} \).

Step 1: Identify the Geometric Series Formula

The formula for the geometric series is \( \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n \), where \(|r| < 1\). In this case, we want to express \(f(x) = \frac{x^{3}}{2-x^{3}}\) in a form similar to \( \frac{1}{1 - r} \).

Step 2: Rewrite the Function

Consider \( f(x) = \frac{x^3}{2-x^3} \). Rewrite it as \( \frac{1}{2-x^3} \) and factor out the 2 to fit the geometric series template, leading to: \( \frac{1}{2} \times \frac{1}{1 - \frac{x^3}{2}} \).

Step 3: Apply the Geometric Series Formula

Using the geometric series formula, express \( \frac{1}{1 - \frac{x^3}{2}} \) as a power series: \( \sum_{n=0}^{\infty} \left(\frac{x^3}{2}\right)^n \). This becomes \( \sum_{n=0}^{\infty} \frac{x^{3n}}{2^n} \).

Step 4: Multiply by Initial Factor

Multiply the entire series by \( \frac{1}{2} \) to adjust for the initial factor taken out, resulting in \( \frac{1}{2} \times \sum_{n=0}^{\infty} \frac{x^{3n}}{2^n} = \sum_{n=0}^{\infty} \frac{x^{3n}}{2^{n+1}} \).

Step 5: Determine the Radius of Convergence

The geometric series converges if \(|r| < 1\). In this case, \( r = \frac{x^3}{2} \), so \( \left| \frac{x^3}{2} \right| < 1 \). Solving for \(x\), we have \( |x^3| < 2 \), or \( |x| < \sqrt[3]{2} \). Thus, the radius of convergence is \( \sqrt[3]{2} \).

Key Concepts

Geometric Series

A geometric series is a sum of the terms in a sequence where each term is a multiple of the previous one by a fixed number, called the common ratio, denoted as \( r \). The infinite geometric series can be expressed by the formula \( \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n \). This series converges when the absolute value of the common ratio \( |r| \) is less than 1.

Understanding this concept is crucial when finding power series representations. If you view the given function \( f(x) = \frac{x^3}{2-x^3} \), it can be manipulated to match the general geometric series formula. By rewriting it as \( \frac{1}{2} \times \frac{1}{1 - \frac{x^3}{2}} \), we see a clear resemblance to the geometric series expression. This allows us to apply the formula to express the function as a power series.

• The common ratio is key when using a geometric series for power series representation.
• Ensure that the expression aligns with \( \frac{1}{1 - r} \) for direct application.

Radius of Convergence

The concept of radius of convergence is essential when dealing with power series. It tells us the range of \( x \) values for which the series converges. For a geometric series of the form \( \sum_{n=0}^{\infty} r^n \), the requirement for convergence is \( |r| < 1 \).

In the context of the given function \( f(x) = \frac{x^3}{2-x^3} \), this translates to finding the range \( x \) values for which the series \( \sum_{n=0}^{\infty} \left(\frac{x^3}{2}\right)^n \) converges. Here, the ratio \( r = \frac{x^3}{2} \), and to satisfy the convergence condition \( \left|\frac{x^3}{2}\right| < 1 \), we solve \( |x^3| < 2 \).

• The radius of convergence, therefore, is \( \sqrt[3]{2} \).
• This means the series converges for \( |x| < \sqrt[3]{2} \).

Understanding the radius of convergence helps in determining how far we can extend our power series approximation around the center of expansion.

Convergence of Series

Convergence of a series is a critical concept in calculus and analysis, signifying whether a series approaches a finite limit as more terms are added. For a power series, convergence is generally concerned with the values of \( x \) that make the series sum to a finite number.

In particular, for the geometric series representation \( \sum_{n=0}^{\infty} \left(\frac{x^3}{2}\right)^n \), we determine convergence based on the radius calculated as \( |x| < \sqrt[3]{2} \). This specific condition means as long as \( x \) stays within \( -\sqrt[3]{2} \) and \( +\sqrt[3]{2} \), the series will converge to a finite value.

• Convergence of series ensures a valid power series approximation only within a certain range of \( x \).
• Understanding its convergence can help in practical applications such as estimating functions in engineering and physics.

By knowing the convergence of the given series, we ascertain where our power series approximation holds true and where it doesn't.