Problem 7 - Applied Discrete Structures

Question

Write as an extended power series: (a) \(\left(x^{4}-x^{5}\right)^{-1}\) (b) \(\left(x^{2}-2 x^{3}+x^{4}\right)^{-1}\)

Step-by-Step Solution

Answer

(a) \((x^4 - x^5)^{-1} = x^{-4} + x^{-3} + \cdots\); (b) \((x^2 - 2x^3 + x^4)^{-1} = x^{-2} + 2x^{-1} + \cdots\).

Step 1: Start with the Geometric Series Formula

Recall the formula for the geometric series: \( \frac{1}{1-a} = 1 + a + a^2 + a^3 + \cdots \). We will use this formula to write each function as a power series.

Step 2: Transform Expression (a)

Rewrite \( (x^4 - x^5)^{-1} \) in the form of the geometric series. Factor out \( x^4 \):\[(x^4(1-x))^{-1} = \frac{1}{x^4} \cdot \frac{1}{1-x}\]Apply the geometric series formula where \( a = x \) thus:\[(1-x)^{-1} = 1 + x + x^2 + x^3 + \cdots\]Multiply by \( x^{-4} \):\[x^{-4} (1 + x + x^2 + x^3 + \cdots) = x^{-4} + x^{-3} + x^{-2} + x^{-1} + 1 + x + \cdots\]

Step 3: Transform Expression (b)

Rewrite \( (x^2 - 2x^3 + x^4)^{-1} \) as a geometric series. Factor \( x^2 \) from the expression:\[(x^2(1-2x+x^2))^{-1} = \frac{1}{x^2} \cdot \frac{1}{1-2x+x^2}\]The term \( 1-2x+x^2 \) can be seen as a perfect square: \( (1-x)^2 \). Since the geometric series formula requires \( |a| < 1 \), and assuming \( |x| < 1 \), this becomes a geometric series when recalculated:\[(1-x)^{-2} = (1-x)^{-1} \cdot (1-x)^{-1} = \left( 1 + x + x^2 + \cdots \right) \left( 1 + x + x^2 + \cdots \right)\]Expanding yields an infinite series. Only terms that can be explicitly expressed in this exercise when initially stated will be visible:

Step 4: Construct Power Series for (b)

Combine the expansions to write the complete power series for \( (x^2 - 2x^3 + x^4)^{-1} \):- \( (1-x)^{-2} = \sum_{n=0}^{\infty} (n+1)x^n \), adjust the index based on multiplying by \( x^{-2} \):\[\frac{1}{x^2}(1 + 2x + 3x^2 + 4x^3 + \cdots) = x^{-2} + 2x^{-1} + 3 + 4x + \cdots\]

Step 5: Finalize the Power Series Expressions

Summarize:(a) \( (x^4 - x^5)^{-1} = x^{-4} + x^{-3} + x^{-2} + x^{-1} + 1 + x + \cdots \)(b) \( (x^2 - 2x^3 + x^4)^{-1} = x^{-2} + 2x^{-1} + 3 + 4x + \cdots \)

Key Concepts

Power Series

A power series is a series of the form \(\sum_{n=0}^{\infty} a_n x^n\), where \(a_n\) represents the coefficients of the series and \(x\) is a variable. It's essentially an infinite polynomial that can represent a wide range of functions. A common use of power series is to approximate functions in a simpler form for calculations.
A power series becomes extremely useful in calculus and mathematical analysis as it allows complex functions to be expressed in simpler forms, enabling easier integration and differentiation.

• Each term in the series involves increasing powers of \(x\).
• The radius of convergence is crucial, dictating where the series is valid and converges to the function it represents.
• The geometric series, a special type of power series, focuses on constant ratios between terms, which serves numerous mathematical applications.

Factoring Expressions

Factoring expressions simplifies mathematical equations, making them easier to handle, especially in algebra and calculus. Factoring is the process of breaking down an expression into products of simpler expressions. It involves recognizing patterns and using common techniques to factorize, such as grouping, using the distributive property, or recognizing special products like squares and cubes.
In the context of the given exercise, factoring was the initial step to transform the complex expressions into a format that aligns directly with the geometric series formula. For instance,

• Factoring out a common term, like in \((x^4 - x^5)^{-1}\), where \(x^4\) is factored out.
• Seeing \(1-2x+x^2\) as a perfect square \((1-x)^2\).
• This simplification aids in applying the geometric series formula to each term separately.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. In mathematics, infinite series often describe numbers or functions by summing infinitely many terms that approach a limit. This concept is fundamental in many fields, showcasing how infinite behavior can effectively represent finite quantities.
For example, when expanding a geometric series, you engage an infinite series involving terms that continue indefinitely. The convergence of such a series defines whether the series sums to a particular value.

• Infinite series are essential in calculus, particularly in methods involving approximation and convergence.
• The sum of an infinite series is computed by considering the limit of partial sums.
• Understanding how and when an infinite series converges is crucial in applications like signal processing and solving differential equations.
• In practical scenarios, often only the first few terms are calculated for approximation.

Geometric Series Formula

The geometric series formula is defined as \(\frac{1}{1-a} = 1 + a + a^2 + a^3 + \cdots\). It describes a series where each term is a constant multiple of the previous term. This formula is particularly powerful due to its simplicity and wide applicability.
For the exercises given, it was employed to expand functions into power series by manipulating expressions into a suitable form such as \(\frac{1}{1-x}\). By transforming complex expressions through factoring, they align into a geometric series framework.

• The formula is a cornerstone in algebra and calculus because it provides a straightforward way to express and handle infinite sequences.
• Geometric series are often used in financial calculations, physics, and computer science for efficient modeling.
• Understanding the conditions for the convergence of a geometric series, typically \(|a|<1\), is essential to ensure valid expansion.