Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The power series is \( \sum_{n=0}^{\infty} \frac{1}{2} \left( \frac{3}{2} \right)^n x^n \), with a radius of convergence \( \frac{2}{3} \).

Step 1: Identifying the Function as a Geometric Series

To represent the function as a geometric series, recognize that this is in the form of a geometric series. The function can be rewritten as \( f(x) = \frac{1}{2} \cdot \frac{1}{1 - \frac{3}{2}x} \), similar to the geometric series formula \( \frac{1}{1-r} = 1 + r + r^2 + \cdots \), for \( |r| < 1 \). Here, \( r = \frac{3}{2}x \).

Step 2: Writing the Power Series

Since \( \frac{1}{1-r} = 1 + r + r^2 + \cdots \), we can expand \( \frac{1}{1 - \frac{3}{2}x} \) using the geometric series formula as:\[1 + \frac{3}{2}x + \left(\frac{3}{2}x\right)^2 + \left(\frac{3}{2}x\right)^3 + \cdots\].Now, multiply each term by the constant factor \( \frac{1}{2} \):\[\frac{1}{2} + \frac{3}{4}x + \frac{9}{8}x^2 + \frac{27}{16}x^3 + \cdots\].

Step 3: Writing the General Term

From the expanded series, observe the pattern of the coefficients. For the \( n \)-th term, the general term is obtained by \( \frac{1}{2} \left( \frac{3}{2} x \right)^n \). So, the series can be written as:\[\sum_{n=0}^{\infty} \frac{1}{2} \cdot \left( \frac{3}{2} \right)^n x^n\].

Step 4: Identifying the Radius of Convergence

The power series, derived from the geometric series formula, converges when the absolute value of the common ratio \( r \) is less than 1. Thus, for convergence:\[ \left| \frac{3}{2} x \right| < 1 \|x| < \frac{2}{3}.\]Thus, the radius of convergence \( R \) is \( \frac{2}{3} \).

Key Concepts

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. You might think of it as a kind of reproductive pattern where each term breeds the next. In mathematical terms, this series can be expressed as:

• First Term: 1
• Recursive Term: Multiply the common ratio (r) with previous term
• General Form: \( 1 + r + r^2 + r^3 + \cdots \)

The beauty of a geometric series is its simplicity and predictability. Notably, the sum of the series \( \frac{1}{1-r} \) takes form when \(|r| < 1\). This clarity allows us to express complex functions such as \( f(x) = \frac{1}{2} \cdot \frac{1}{1 - \frac{3}{2}x} \) by quickly determining the power series by observation. We transform the function into a power series by considering each term as a part of the geometric progression.

Radius of Convergence

Understanding the radius of convergence is pivotal when working with power series. Simply put, the radius of convergence defines the distance from the center of convergence where the series converges.

• The center is typically given as a point on the x-axis—here, at x = 0 for simplicity.
• The convergence occurs within a radius \( R \) around the center.
• Outside this radius, the series diverges or does not sum up to a finite number.

In the case where our function \( f(x) = \frac{1}{2 - 3x} \), we expressed it as a power series through a geometric series. The radius of convergence is determined by ensuring the absolute value of the ratio \( \frac{3}{2}x \) is less than 1. This gives us the condition \( |x| < \frac{2}{3} \). Thus, the radius of convergence is \( R = \frac{2}{3} \), indicating the power series sums within this interval centered at x = 0.

Convergence of Series

Convergence is a term that signifies whether the sum of an infinite series approaches a finite limit as the number of terms increases. Not all series converge, and understanding convergence is crucial when working with series in calculus and analysis.

• A series converges if the sequence of partial sums approaches a specific finite number.
• In a geometric series like \( 1 + r + r^2 + \ldots \), if \(|r| < 1\), the series converges to \( \frac{1}{1-r} \).
• If \(|r| \geq 1\), the series does not have a finite sum and thus diverges.

For our function \( f(x) = \frac{1}{2 - 3x} \), we applied the condition for convergence from a geometric perspective. We continue to expand it into a power series, knowing it converges in the interval where the common ratio satisfies \(|r| < 1\). This provides meaningful insights into the behavior of functions expressed in this form and informs us about their limits and possible applications within bounds of convergence.