Problem 5 - Applied Discrete Structures

Question

(a) Determine the inverse of \(h(x)=\sum_{i=0}^{\infty} 2^{i} x^{i}\) in \(\mathbb{Q}[[x]]\). (b) Use the procedures in Chapter 8 to find a rational generating function for \(h(x)\) in part a. Find the multiplicative inverse of this function.

Step-by-Step Solution

Answer

The inverse of \( h(x) \) is \( 1 - 2x \). The rational generating function is \( \frac{1}{1-2x} \).

Step 1: Understanding the Series

The function given is a power series: \[ h(x) = \sum_{i=0}^{\infty} 2^{i} x^{i} = 1 + 2x + 4x^2 + 8x^3 + \cdots \]This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = 2x \).

Step 2: Sum of the Geometric Series

We use the formula for the sum of an infinite geometric series, \( \sum_{i=0}^{\infty} ar^i = \frac{a}{1-r} \), where \(|r| < 1\) to find\[ h(x) = \frac{1}{1 - 2x} \]Thus, the generating function for \( h(x) \) is the rational function \( \frac{1}{1 - 2x} \).

Step 3: Find the Multiplicative Inverse

The multiplicative inverse of the series \( h(x) \) in \( \mathbb{Q}[[x]] \) is the inverse of the rational function. Since \( h(x) = \frac{1}{1-2x} \), its inverse is simply \[ (1 - 2x) \].This is because multiplying \( \frac{1}{1-2x} \times (1-2x) = 1 \).

Key Concepts

Geometric Series

In mathematics, a geometric series is a sum of terms, each of which is a constant multiple of the previous one. It's important because it gives us a simple way to represent repeated multiplications as sums. A geometric series has the general form: - \( a + ar + ar^2 + ar^3 + \cdots \), where:

• \( a \) is the first term, and
• \( r \) is the common ratio.

For an infinite series, like in our exercise, there is an elegant formula to find the sum: \[ \sum_{i=0}^{\infty} ar^i = \frac{a}{1 - r} \] provided \( |r| < 1 \). This ensures that the series converges to a finite sum. In our case, the series \( h(x) = \sum_{i=0}^{\infty} 2^i x^i \) is a geometric series with \( a = 1 \) and \( r = 2x \). As long as \( |2x| < 1 \), or \(|x| < \frac{1}{2}\), the sum of this infinite series is \( \frac{1}{1 - 2x} \). This is a very neat way of expressing complicated combinations of terms as a single simple fraction.

Generating Functions

Generating functions are powerful tools used in combinatorics and other areas of math for counting and handling sequences. A generating function is like a "data set in disguise" that uses a power series to encode information about a sequence of numbers. For our series, the generating function is the power series itself, \( h(x) = \sum_{i=0}^{\infty} 2^i x^i \). This is not just a random series; it represents a structured way of organizing coefficients that can be manipulated.What's fascinating about using generating functions is that we can convert the series into a rational function using the geometric series formula. For instance, in this exercise, we turned the series into its closed form \( \frac{1}{1 - 2x} \). This transformed function can be used to find various properties about the sequence, such as sums, averages, and much more, making analysis considerably simpler.

Multiplicative Inverse

The concept of a multiplicative inverse is fundamental in mathematics, as it refers to a number that, when multiplied with a given number, results in one. For a function or number \( a \), the multiplicative inverse is \( b \) if \( a \cdot b = 1 \).In the context of power series and rational functions, finding the multiplicative inverse is crucial when manipulating these functions, especially in fields like calculus and algebra.In our textbook solution, we determined that the inverse of the geometric series \( h(x) = \frac{1}{1-2x} \) is simply \( (1-2x) \). We know this because multiplying these two gives us 1:

• \( \frac{1}{1-2x} \times (1-2x) = 1 \)

This basic understanding of inverses helps us solve equations involving rational functions and series, as it allows us to "undo" or "reverse" a function, especially when solving for unknown values.