Problem 1 - Applied Discrete Structures

Question

\begin{aligned} &\text { Let } f(x)=\sum_{i=0}^{\infty} a_{i} x^{i} \text { and } g(x)=\sum_{i=0}^{\infty} b_{i} x^{i} \text { be elements of } R[[x]] . \text { Let }\\\ &f(x) \cdot g(x)=\sum_{i=0}^{\infty} d_{i} x^{i}=1 . \text { Apply basic algebra to }\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots\right)\\\ &\left(b_{0}+b_{1} x+b_{2} x^{2}+\cdots\right) \text { to derive the formula } d_{s}=\sum_{i=0}^{s} a_{i} b_{s-i} \text { for the co- }\\\ &\text { efficients of } f(x) \cdot g(x) \text { . Hence, to show that } f(x) \cdot g(x)=\sum_{s=0}^{\infty}\left(\sum_{i=0}^{s} a_{i} b_{s-i}\right) x^{s} \end{aligned}

Step-by-Step Solution

Answer

The coefficients are given by \( d_s = \sum_{i=0}^{s} a_i b_{s-i} \) for \( f(x) \cdot g(x) = 1 \).

Step 1: Understanding the Problem

We are given two power series functions, \( f(x) \) and \( g(x) \), where each function is expressed as a sum of infinite terms with coefficients \( a_i \) and \( b_i \), respectively. Our goal is to find a formula for the coefficients, \( d_s \), of the product \( f(x) \cdot g(x) \), which equals \( 1 \).

Step 2: Setting Up the Multiplication

Consider the expansion of the product of two power series: \( f(x) = a_0 + a_1 x + a_2 x^2 + \, \cdots \) and \( g(x) = b_0 + b_1 x + b_2 x^2 + \, \cdots \). The product \( f(x) \cdot g(x) \) is expressed as \( (a_0 + a_1 x + a_2 x^2 + \, \cdots)(b_0 + b_1 x + b_2 x^2 + \, \cdots) \).

Step 3: Applying Infinite Series Multiplication

Multiply each term from \( f(x) \) by each term from \( g(x) \) and collect terms of the same power of \( x \). This yields the general form \( d_s x^s = (\sum_{i=0}^{s} a_i b_{s-i}) x^s \), where each \( d_s \) is the coefficient of \( x^s \) in the expanded product series.

Step 4: Understanding the Coefficients

The coefficients \( d_s \) are formed by taking the sum of products of coefficients from \( f(x) \) and \( g(x) \) such that their indices add up to \( s \). Thus, \( d_s = \sum_{i=0}^{s} a_i b_{s-i} \). This represents the standard method for finding the coefficient of a power series product.

Step 5: Verifying the Formula

From the series form, confirm that \( f(x) \cdot g(x) = \sum_{s=0}^{\infty}\left(\sum_{i=0}^{s} a_i b_{s-i}\right) x^s \). This equality confirms that the derived formula for \( d_s \) satisfies the relation for each power of \( x \). When the product equals 1, \( d_0 = 1 \), and all other \( d_s = 0 \) for \( s > 0 \).

Key Concepts

Coefficient Formula

The coefficient formula is an important concept when working with power series. When you have two power series like \( f(x) = \sum_{i=0}^{\infty} a_{i} x^{i} \) and \( g(x) = \sum_{i=0}^{\infty} b_{i} x^{i} \), their product produces another power series. This resulting series is expressed as \( f(x) \cdot g(x) = \sum_{i=0}^{\infty} d_{i} x^{i} \). To find the coefficients \( d_s \) in this product series, we use the formula:

• \( d_s = \sum_{i=0}^{s} a_{i} b_{s-i} \)

This formula is derived by taking the sum of products of coefficients \( a_i \) and \( b_{s-i} \) such that their indices add up to \( s \). It allows us to build each coefficient of the resulting power series from the combinations of terms in the original series. It's a crucial mechanism for understanding how complex functions break down into simpler, manageable components.

Infinite Series

An infinite series is a sum of infinite terms. In the context of power series like \( f(x) = \sum_{i=0}^{\infty} a_{i} x^{i} \), each term is a function of \( x \) raised to increasing powers, multiplied by coefficients. Infinite series are widely used in mathematical analysis because they allow us to express functions as sums of simpler terms. When multiplying two infinite series, the challenge is dealing with infinitely many terms and finding meaningful sums that resolve to specific coefficients. Despite involving complex calculations, the beauty of infinite series is their ability to approximate functions over large intervals, often leading to closed-form solutions or convergent power series that can have practical applications in physics, engineering, and beyond.

Algebraic Manipulation

Algebraic manipulation is the process of modifying an expression into a different form using the rules of algebra. In the context of power series, algebraic manipulation involves expanding and simplifying expressions, which leads us to determine specific coefficients in series multiplications. For this exercise, it involves organizing the product \((a_0 + a_1 x + a_2 x^2 + \cdots)(b_0 + b_1 x + b_2 x^2 + \cdots)\) into a form where each \(x^s\) has a clear coefficient \(d_s\). By systematically arranging and simplifying terms, you can see which coefficients contribute to each power of \(x\), helping you apply the coefficient formula effectively. Mastering these manipulations provides a foundation for more complex algebraic and calculus operations that are crucial in higher-level mathematics.

Power Series Multiplication

Power series multiplication involves multiplying two sums of series to find a new series as their product. The goal is to discover a pattern or formula that describes how individual terms of the two series interact to form a new sequence.In our scenario, multiplying \(f(x)\) and \(g(x)\) requires taking every term in \(f(x)\) and multiplying it by every term in \(g(x)\). The product forms mixed power terms that, when combined, create a new power series. Each term in the resulting series can be organized by their power of \(x\), with each power having its coefficient as a sum of products of original series coefficients. The multiplication is systematic and orderly, following the principle of the distributive property in algebra. Understanding power series multiplication is critical in contexts where functions are composed, such as solving differential equations and evaluating complex integrations.