Problem 14 - Calculus: One and Several Variables

Question

Expand \(f(x)\) in powers of \(x\) $$f(x)=\frac{1-x}{1+x}$$

Step-by-Step Solution

Answer

The expanded form of the function \(f(x) = \frac{1-x}{1+x}\) in powers of \(x\) is \(f(x) = 1 - x - x^2 - x^3 - x^4 \ldots\).

Step 1: Rewrite the function using the geometric series formula

We will rewrite the function \(f(x)\) using the geometric series formula. Let's substitute \(r\) with \(-x\), since we see the "division" by \(1+x\): $$f(x) = 1 - x (1 - (-x) + (-x)^2 - (-x)^3 + \ldots )$$

Step 2: Distribute the negative sign

Next, we will distribute the negative sign to the terms inside the parentheses: $$f(x) = 1 - x (1 + x + x^2 + x^3 + \ldots )$$

Step 3: Distribute the x

Now, distribute \(x\) to each term in the infinite series: $$f(x) = 1 - (x + x^2 + x^3 + x^4 + \ldots )$$

Step 4: Write the final result

Finally, let's write the result as an expanded form of powers of \(x\): $$f(x) = 1 - x - x^2 - x^3 - x^4 \ldots$$ Thus, the expanded form of the function \(f(x) = \frac{1-x}{1+x}\) in powers of \(x\) is \(f(x) = 1 - x - x^2 - x^3 - x^4 \ldots\).

Key Concepts

Geometric Series

The geometric series is a series of numbers in which each term is a fixed multiple of the previous term. This multiplier is commonly referred to as the "common ratio." A geometric series can be either finite or infinite. An infinite geometric series takes the form:

• \[ S = a + ar + ar^2 + ar^3 + ext{...} \]

where \( a \) is the first term, and \( r \) is the common ratio. The formula for the sum of an infinite geometric series, when the absolute value of the common ratio \( |r| < 1 \), is \( S = \frac{a}{1-r} \). This powerful tool lets mathematicians represent functions as infinite sums, allowing for easier computation and analysis.

In the context of the exercise, rewriting the function \( f(x) = \frac{1-x}{1+x} \) using a geometric series provides an expansion in powers of \( x \). Recognizing this pattern helps streamline complex functions by breaking them into simpler components.

Power Series

A power series is an infinite series of the form:

• \[ f(x) = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + ext{...} \]

where \( a_n \) are coefficients and \( c \) is the center of the series. Power series are a way to express complicated functions as sums of simpler polynomial terms. They provide great flexibility in mathematical manipulation and are a core concept in calculus and analysis.

The expanded form \( f(x) = 1 - x - x^2 - x^3 - x^4 \ldots \) is an example of a power series where the coefficients are all -1 except for the constant term, which is 1, and is centered at \( c = 0 \). Each term is a power of the variable, revealing the function's behavior for small values of \( x \). This is invaluable in approximating functions.

Infinite Series

An infinite series is a sum of an infinite sequence of numbers. Each number in the series is called a term, and the series can be expressed as:

• \[ S = a_1 + a_2 + a_3 + ext{...} = \sum_{n=1}^{\infty} a_n \]

Infinite series can converge, leading to a specific value, or diverge. Convergence implies the series approaches a certain number as more terms are added. This is determined by various tests, like the ratio test and the root test.

In the solution for the function \( f(x) \), the step-by-step approach transformed it into an infinite series. Each term \( -x^n \) gradually adds more detail to the function's behavior, illustrating how infinite series allow us to describe more complicated functions effectively and compactly.

Function Expansion

Function expansion is the process of expressing a function in terms of a series or sum, usually turning a complex expression into more straightforward components. This is a powerful method for analyzing functions and is heavily used in calculus.

• It makes functions easier to differentiate and integrate.
• It helps in solving differential equations and evaluating limits.
• It provides insights into the behavior of functions near certain points.

Function expansion often involves using techniques like Taylor series or Maclaurin series, which expand functions about a point. For \( f(x) = \frac{1-x}{1+x} \), using geometric and power series expansion made handling the original fraction straightforward by distributing terms of the infinite series. Such transformations reveal underlying patterns and simplify computations.