Problem 2 - Applied Discrete Structures

Question

What sequences have the following generating functions? (a) \(\frac{1}{1+z}\) (b) \(\frac{1}{4-3 z}\) (c) \(\frac{2}{1-2}+\frac{1}{1+z}\) (d) \(\frac{z+2}{z+3}\)

Step-by-Step Solution

Answer

(a) \(1, -1, 1, -1, \ldots\); (b) \(\frac{1}{4}, \frac{3}{16}, \ldots\); (c) \(-2,-1,-3, \ldots\); (d) \(1, -\frac{1}{3}, \ldots\)

Step 1: Identify the Formula for (a)

The generating function given is \( \frac{1}{1+z} \). Recognize this as a geometric series \( \sum_{n=0}^{\infty} (-1)^n z^n \) based on the formula \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \). Substitute \( x \) with \( -z \).

Step 2: Calculate Sequence for (a)

Using the formula identified, the sequence corresponding to \( \frac{1}{1+z} \) is a geometric series where each term is \((-1)^n\). Therefore, the sequence is \(1, -1, 1, -1, 1, \ldots\)

Step 3: Identify the Formula for (b)

The generating function is \( \frac{1}{4-3z} \). This is a geometric series of the form \( \sum_{n=0}^{\infty} a_n z^n \). First rewrite the expression as \( \frac{1}{4} \cdot \frac{1}{1 - \frac{3z}{4}} \). Then recognize the second term as a geometric series \( \sum_{n=0}^{\infty} \left( \frac{3}{4} \right)^n z^n \).

Step 4: Calculate Sequence for (b)

Using the rewritten expression, obtain the sequence coefficients \( a_n = \left( \frac{3}{4} \right)^n \) scaled by the factor \( \frac{1}{4} \). Thus, the sequence is \( \frac{1}{4}, \frac{3}{16}, \frac{9}{64}, \ldots \).

Step 5: Identify the Formula for (c)

The function is given by \( \frac{2}{1-2} + \frac{1}{1+z} \). The first term is a constant \( -2 \) because \( \frac{2}{1-2} = -2 \). The second term is similar to part (a), which we found to be \( \sum_{n=0}^{\infty} (-1)^n z^n \).

Step 6: Calculate Sequence for (c)

The sequence is a combination of a constant \( -2 \) and the sequence from the \( \frac{1}{1+z} \) generating function. Thus, it is: \(-2 , -2+1, -2+(-1), -2+1, \ldots\), which simplifies to: -2, -1, -3, -1, -3, \ldots\.

Step 7: Simplify the Expression for (d)

The function \( \frac{z+2}{z+3} \) cannot be simplified easily into a geometric series form. Perform long division for \( \frac{z+2}{z+3} \) which gives \(1 - \frac{1}{z+3}\). Calculate the remaining term \( - \frac{1}{z+3} = \sum_{n=0}^{\infty} (-1)^{n+1} 3^{-n-1} z^n \).

Step 8: Calculate Sequence for (d)

Using the series expansion: the sequence is \(1, -\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \ldots\) after combining \(1\) from long division and the series expansion.

Key Concepts

Geometric Series

A geometric series is a type of series in mathematics where each term is a constant multiple of the previous term. This repeated multiplication by a fixed number, known as the ratio, forms the backbone of the geometric series. A simple example is

• When you see the expression \( \frac{1}{1-x} \), you're looking at a sum of the form \( \sum_{n=0}^{\infty} x^n \).
• This series converges when the absolute value of \(x\) is less than 1.

Such series are essential in calculating or estimating values that inherently follow a repeated multiplication process, like interest growth or decay in physics. Understanding this concept allows us to manipulate and use generating functions effectively.

Sequence Calculations

In mathematics, sequences are ordered lists of numbers created by a specific rule or pattern. Sequence calculations often involve finding a general formula that describes the nth term in the sequence. Here’s how that works:

• Given a generating function, like \( \frac{1}{1+z} \), we identify it as a type of geometric series.
• Once identified, we know it follows the pattern \( (-1)^n \) in terms of the power of \( z \), creating the sequence 1, -1, 1, -1, ...

In this way, generating functions are pivotal tools, providing a shorthand expression to performance analysis and calculations of infinitely repeating patterns or sequences.

Geometric Series Expansion

Geometric series expansions are expressions of functions as infinite series sums. These expansions allow decomposing complex fractions into sum forms that are easier to understand and work with. Let’s take a closer look:

• To rewrite \( \frac{1}{4-3z} \), you decompose it into \( \frac{1}{4} \times \frac{1}{1-\frac{3z}{4}} \).
• The latter expression is recognized as a geometric series \( \sum_{n=0}^{\infty} \left(\frac{3}{4}\right)^n z^n \).

This transformation is key for applications in areas like signal processing and complex computations. It ensures that complex functions remain manageable for further work, revealing patterns and predictions inherent in the series.

Series Coefficients

When working with series, coefficients describe the contribution of each term to the series sum. In essence:

• From the generating function, coefficients \(a_n\) are calculated directly from the expansion steps.
• For \( \frac{1}{4-3z} \), coefficients scale by \( \frac{1}{4} \), resulting in the sequence \( \frac{1}{4}, \frac{3}{16}, \frac{9}{64}, \ldots \).
• Each term's coefficient, \( \left(\frac{3}{4}\right)^n \), reflects accumulated changes, like growth rates or frequencies.

Through understanding series coefficients, we distribute value across parts of a function, revealing structure and magnitude that's instrumental in mathematical modeling.