Question

Find the first five terms of the Maclaurin series (i.e., choose \(n=4\) and let \(x_{0}=0\) ) for: (a) \(\phi(x)=\frac{1}{1-x}\) (b) \(\phi(x)=\frac{1-x}{1+x}\)

Short answer

(a) Maclaurin series: \(1 + x + x^2 + x^3 + x^4\). (b) Maclaurin series: \(1 - 2x + 2x^2 - 2x^3 + 2x^4\).

Step-by-step solution

Understand the Maclaurin Series Formula

The Maclaurin series is a type of Taylor series centered at zero (\(x_0 = 0\)). It is given by the formula: \[ \phi(x) = \phi(0) + \phi'(0)x + \frac{\phi''(0)}{2!}x^2 + \frac{\phi'''(0)}{3!}x^3 + \ldots \]To find the first five terms, we will compute up to the fourth derivative of each function and evaluate them at \(x=0\).

Compute Derivatives for (a)

For \( \phi(x) = \frac{1}{1-x} \):1. \( \phi'(x) = \frac{1}{(1-x)^2} \)2. \( \phi''(x) = \frac{2}{(1-x)^3} \)3. \( \phi'''(x) = \frac{6}{(1-x)^4} \)4. \( \phi^{(4)}(x) = \frac{24}{(1-x)^5} \)

Evaluate Derivatives at x=0 for (a)

Evaluate the derivatives at \(x=0\):1. \( \phi(0) = 1 \)2. \( \phi'(0) = 1 \)3. \( \phi''(0) = 2 \)4. \( \phi'''(0) = 6 \)5. \( \phi^{(4)}(0) = 24 \)

Write the Maclaurin Series for (a)

Substitute the values into the Maclaurin series formula:\[ \phi(x) = 1 + x + \frac{2}{2}x^2 + \frac{6}{6}x^3 + \frac{24}{24}x^4 \]Simplify to:\[ \phi(x) = 1 + x + x^2 + x^3 + x^4 \]

Compute Derivatives for (b)

For \( \phi(x) = \frac{1-x}{1+x} \):1. \( \phi'(x) = \frac{-2}{(1+x)^2} \)2. \( \phi''(x) = \frac{4}{(1+x)^3} \)3. \( \phi'''(x) = \frac{-12}{(1+x)^4} \)4. \( \phi^{(4)}(x) = \frac{48}{(1+x)^5} \)

Evaluate Derivatives at x=0 for (b)

Evaluate the derivatives at \(x=0\):1. \( \phi(0) = 1 \)2. \( \phi'(0) = -2 \)3. \( \phi''(0) = 4 \)4. \( \phi'''(0) = -12 \)5. \( \phi^{(4)}(0) = 48 \)

Write the Maclaurin Series for (b)

Substitute the values into the Maclaurin series formula:\[ \phi(x) = 1 - 2x + \frac{4}{2}x^2 - \frac{12}{6}x^3 + \frac{48}{24}x^4 \]Simplify to:\[ \phi(x) = 1 - 2x + 2x^2 - 2x^3 + 2x^4 \]

Key concepts

Taylor Series

The Taylor Series is a fundamental concept in calculus that allows us to represent complex functions as infinite sums of simpler polynomials. This method is incredibly powerful because it simplifies analysis and calculations involving functions that may initially be difficult to handle. The Taylor Series can be expanded at any point, but with the Maclaurin Series, we specifically center the expansion at zero. Such expansions are used to approximate functions in various scientific and engineering problems.

Mathematically, the Taylor Series of a function \( f(x) \) at a point \( x_0 \) is given by:

• \( f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \ldots \)

For the Maclaurin Series, we replace \( x_0 \) with zero and the formula simplifies. Understanding how Taylor Series are constructed is essential for modeling and approximating real-world phenomena, as it breaks down otherwise complicated functions into more manageable parts.

Derivative Evaluation

Evaluating derivatives is a critical step in constructing both Taylor and Maclaurin Series. Each derivative provides a building block for approximating functions. Derivative evaluation is the process of calculating the rate at which a function changes at any given point.

In our exercise, we calculated the derivatives of \( \phi(x) = \frac{1}{1-x} \) and \( \phi(x) = \frac{1-x}{1+x} \), each up to the fourth order. These derivatives allow us to construct the series terms:

• The first derivative measures the slope or rate of change at the initial point.
• The second derivative informs us about the curvature or how the rate of change itself is changing.
• Higher-order derivatives continue to provide more detailed insight into the behavior of a function.

By evaluating these derivatives at \( x=0 \), we can use them directly in forming a Maclaurin Series, highlighting their relevance in approximation and analysis.

Series Expansion

Series Expansion is the process of expressing functions as a sum of terms that are easier to compute and understand than the original function. In the context of Taylor and Maclaurin Series, we expand a function into its infinite series form.

The primary goal of series expansion is to approximate a function as closely as possible using a finite number of terms. This comes in handy especially when calculating explicit values or predicting behavior around a point where the function might not be easy to evaluated directly.

• By using the derivatives, we evaluate terms that approximate the function incrementally.
• We typically use only the first few terms for practical calculations, as they already provide a significant level of accuracy.

In our example, the first five terms involve the application of derivatives up to the fourth order. This series expansion reveals the nuances of a function’s behavior near \( x=0 \), making it a powerful tool in computational mathematics.

Mathematical Economics

In Mathematical Economics, functions and their approximations play critical roles, particularly when analyzing trends and forecasting economic indicators. The Maclaurin Series, a type of Taylor Series, becomes invaluable in this field for breaking down complex economic models into simpler components for approximation and analysis.

The ability to simplify models using series allows economists to make precise predictions without solving the full complexity of an economic model. Economists often use these series to:

• Approximate consumer demand or supply functions around an equilibrium point.
• Forecast economic growth or decline using calculated derivative data.
• Analyze stability by using the model’s derivatives to predict behavior around certain economic thresholds.

This approximation by series provides economists with a practical way to manage and manipulate the rich data sets that economic models comprise, leading to more efficient decision-making processes.