Problem 6 - Calculus: Early Transcendentals

Question

Find a power series representation for the function and determine the interval of convergence. \( f(x) = \frac {4}{2x + 3} \)

Step-by-Step Solution

Answer

The power series is \( \sum_{n=0}^{\infty} \frac{4}{3} \left(-\frac{2}{3}\right)^n x^n \) with interval of convergence \( -\frac{3}{2} < x < \frac{3}{2} \).

Step 1: Rewrite the Function

First, we rewrite the function in a form that will allow us to eventually express it as a geometric series. Start by factoring out the constant from the denominator:\[ f(x) = \frac{4}{3} \cdot \frac{1}{1 - (-\frac{2x}{3})} \]This allows us to put the function in a form that can be expressed as a geometric series.

Step 2: Identify the Geometric Series

We recall that the geometric series formula is:\[ \sum_{n=0}^{\infty} r^n = \frac{1}{1-r} \]Here, we identify that our function already resembles this form with \( r = -\frac{2x}{3} \). Thus:\[ \frac{1}{1 - (-\frac{2x}{3})} = \sum_{n=0}^{\infty} \left(-\frac{2x}{3}\right)^n \]

Step 3: Construct the Power Series

Now that we have identified the geometric series, we multiply it by the factor we factored out previously:\[ f(x) = \frac{4}{3} \cdot \sum_{n=0}^{\infty} \left(-\frac{2x}{3}\right)^n = \sum_{n=0}^{\infty} \frac{4}{3} \cdot \left(-\frac{2}{3}\right)^n x^n \]Thus, the power series representation for \( f(x) \) is:\[ \sum_{n=0}^{\infty} \frac{4}{3} \left(-\frac{2}{3}\right)^n x^n \]

Step 4: Determine the Interval of Convergence

The interval of convergence can be determined by finding where the series converges. For a geometric series \( \sum_{n=0}^{\infty} r^n \) to converge, the condition is \( |r| < 1 \). Here, \( r = -\frac{2x}{3} \), so:\[ \left| -\frac{2x}{3} \right| < 1 \]Simplifying this inequality gives:\[ \left| \frac{2x}{3} \right| < 1 \]\[ \left| x \right| < \frac{3}{2} \]Thus, the interval of convergence is \( -\frac{3}{2} < x < \frac{3}{2} \).

Key Concepts

Understanding Geometric Series

A geometric series is a type of series where each term is derived by multiplying the previous term by a fixed, non-zero number called the ratio. This fixed number is key in identifying and expressing a series as geometric.
The standard form of a geometric series is \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio.
For example, the series \( 1 + 2 + 4 + 8 + ... \) is a geometric series where \( a = 1 \) and \( r = 2 \).

• If \( |r| < 1 \), the series converges, meaning it adds up to a finite value.
• If \( |r| \ge 1 \), the series diverges, continuing indefinitely without reaching a sum.

Understanding this allows us to transform functions into power series, aiding in mathematical analysis and computation.

Exploring the Interval of Convergence

The interval of convergence of a power series is the range of values for which the series converges, or in other words, sums to a finite value.
This interval is critical in understanding the behavior of functions represented by power series.
To determine an interval of convergence for a geometric series, we focus on the ratio \( r \). The series converges if \( |r| < 1 \).
For our example, we have \( r = -\frac{2x}{3} \), so:
\( \left| \frac{2x}{3} \right| < 1 \) simplifies to \( |x| < \frac{3}{2} \), giving us the interval \( -\frac{3}{2} < x < \frac{3}{2} \). Understanding this can help solve various problems in analysis.

• An interval of convergence can be finite as in the example, or infinite, extending from negative to positive infinity.
• The endpoints of the interval need to be checked individually for convergence.

What Makes a Series Convergent?

A convergent series is one that converges to a specific value. This means that as you add more terms, the series approaches a finite sum.
This concept is central in mathematics, especially in calculus and analysis.
To check if a series converges, we often use tests. The geometric series test is a straightforward method for series with constant ratios.

• If the ratio of each term \( r \) is less than 1 in absolute value, the series converges.
• Otherwise, it diverges, meaning it continues indefinitely without settling on a finite value.

Convergence means stability and predictability in series, making it crucial for functions expressed through power series, like the one in our example with the interval of convergence \( -\frac{3}{2} < x < \frac{3}{2} \). Understanding these concepts is pivotal in mastering series representation and manipulation.