Problem 6 - Applied Discrete Structures

Question

Let \(a(x)=1+3 x+9 x^{2}+27 x^{3}+\cdots=\sum_{i=0}^{\infty} 3^{i} x^{i}\) and \(b(x)=1+x+x^{2}+\). \(x^{3}+\cdots=\sum_{i=0}^{\infty} x^{i}\) both in \(\mathbb{R}[[x]]\). (a) What are the first four terms (counting the constant term as the \(0^{\text {th }}\) term \()\) of \(a(x)+b(x) ?\) (b) Find a closed form expression for \(a(x)\). (c) What are the first four terms of \(a(x) b(x) ?\)

Step-by-Step Solution

Answer

(a) 2 + 4x + 10x^2 + 28x^3; (b) \(\frac{1}{1-3x}\); (c) 1 + 4x + 13x^2 + 40x^3.

Step 1: First four terms of a(x) + b(x)

To find the first four terms of \(a(x) + b(x)\), we start by writing the first four terms of each series:1. For \(a(x)\), the terms are \(1 + 3x + 9x^2 + 27x^3\).2. For \(b(x)\), the terms are \(1 + x + x^2 + x^3\).Now, add the corresponding terms from \(a(x)\) and \(b(x)\) together.- Constant term: \(1 + 1 = 2\)- Linear term: \(3x + x = 4x\)- Quadratic term: \(9x^2 + x^2 = 10x^2\)- Cubic term: \(27x^3 + x^3 = 28x^3\)Thus, the first four terms of \(a(x) + b(x)\) are \(2 + 4x + 10x^2 + 28x^3\).

Step 2: Finding a closed form for a(x)

The function \(a(x)\) is a geometric series with a starting term \(a_0 = 1\) and a common ratio \(r = 3x\). The sum of an infinite geometric series is given by the formula:\[S = \frac{a_0}{1 - r}\]Substituting the values, we get:\[a(x) = \frac{1}{1 - 3x}\]This is the closed form expression for \(a(x)\).

Step 3: First four terms of a(x) * b(x)

To find the first four terms of \(a(x) \cdot b(x)\), consider the product of the series: - The constant term is \(1 \cdot 1 = 1\). - The linear term comes from: \(1 \cdot x + 3x \cdot 1 = 4x\).- The quadratic term is: \(1 \cdot x^2 + 3x \cdot x + 9x^2 \cdot 1 = 13x^2\).- The cubic term is: \(1 \cdot x^3 + 3x \cdot x^2 + 9x^2 \cdot x + 27x^3 \cdot 1 = 40x^3\).Thus, the first four terms of \( a(x) \cdot b(x)\) are \(1 + 4x + 13x^2 + 40x^3 \).

Key Concepts

Closed Form Expression

A closed form expression is a way of expressing an infinite series as a single, neat formula. This is incredibly useful, as it allows us to work with a simple expression instead of cumbersome summations. For the given problem, the function \( a(x) = 1 + 3x + 9x^2 + 27x^3 + \cdots \) represents a geometric series. In a geometric series, each term is obtained by multiplying the previous term by a constant, known as the common ratio. Here, our common ratio \( r \) is \( 3x \).
For a geometric series, the closed form expression can be found using the formula:

• \( S = \frac{a_0}{1-r} \)

where \( a_0 \) is the first term of the series. For our series, \( a_0 \) is 1. By substituting in the values from the problem, we find the closed form:

• \( a(x) = \frac{1}{1-3x} \)

This expression allows us to calculate the sum of the entire infinite series simply and quickly based on the value of \( x \). Closed forms are a powerful tool in mathematics, particularly in handling infinite series efficiently.

Series Addition

Series addition involves summing the corresponding terms of two or more series to form a new series. In the exercise, we look at adding \( a(x) \) and \( b(x) \). Each of these are represented as infinite series of terms involving powers of \( x \).
The first step is to list out the initial terms of each series:

• For \( a(x): 1 + 3x + 9x^2 + 27x^3 + \cdots \)
• For \( b(x): 1 + x + x^2 + x^3 + \cdots \)

To find the sum \( a(x) + b(x) \), you simply add the corresponding terms from each series, term by term:

• Constant term: \( 1 + 1 = 2 \)
• Linear term: \( 3x + x = 4x \)
• Quadratic term: \( 9x^2 + x^2 = 10x^2 \)
• Cubic term: \( 27x^3 + x^3 = 28x^3 \)

The result is a new series \( 2 + 4x + 10x^2 + 28x^3 + \cdots \). Each term in this new series is a sum of the terms of corresponding degrees from the original series.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two series to form a new one. It involves taking each term in one polynomial and multiplying it by every term in the other.
In our case, we are multiplying \( a(x) \) and \( b(x) \):

• \( a(x) = 1 + 3x + 9x^2 + 27x^3 + \cdots \)
• \( b(x) = 1 + x + x^2 + x^3 + \cdots \)

To find the first four terms of \( a(x) \cdot b(x) \), examine the combinations of terms:

• Constant term: \( 1 \cdot 1 = 1 \)
• Linear term: \( 1 \cdot x + 3x \cdot 1 = 4x \)
• Quadratic term: \( 1 \cdot x^2 + 3x \cdot x + 9x^2 \cdot 1 = 13x^2 \)
• Cubic term: \( 1 \cdot x^3 + 3x \cdot x^2 + 9x^2 \cdot x + 27x^3 \cdot 1 = 40x^3 \)

The first four terms of the resulting polynomial are \( 1 + 4x + 13x^2 + 40x^3 \). By multiplying each term in this systematic way, you ensure that every possible contribution to each power of \( x \) is accounted for.

Infinite Series

An infinite series is the sum of the terms in an infinite sequence. Unlike a finite series that has a set number of terms, an infinite series theoretically continues forever. Understanding infinite series is crucial for many areas of mathematics.
In this exercise, both \( a(x) \) and \( b(x) \) are represented as infinite series:

• \( a(x) = \sum_{i=0}^{\infty} 3^{i} x^{i} \)
• \( b(x) = \sum_{i=0}^{\infty} x^{i} \)

These series indicate that each subsequent term within the series involves powers of \( x \) that increase by one unit as the series progresses. Although such series go on indefinitely, only a handful of terms might be significant for practical calculations or approximations in many areas such as physics, engineering, or financial modeling.
Infinite series help us understand complex functions, but working directly with them can be challenging. That's why finding a closed form expression or simplifying the series with initial terms is often useful. Awareness of convergence—a property referring to whether the sum of an infinite series approaches a finite number—is also crucial. In our case, the closed form helps determine when the series is well-defined and when it diverges.