Question

Find the Maclaurin polynomial of degree \(n\) for the given function. $$f(x)=\frac{1}{1-x}, \quad n=4$$

Short answer

The Maclaurin polynomial of degree 4 is: 1 + x + x^2 + x^3 + x^4.

Step-by-step solution

Understand the Maclaurin Series

A Maclaurin series is a special case of the Taylor series centered at zero. It is an infinite series that represents a function as a sum of its derivatives at zero. The Maclaurin series for a function \(f(x)\) is given by: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \cdots \]

Recognize the Template for Geometric Series

The given function \(f(x) = \frac{1}{1-x}\) has the geometric series representation: \[ f(x) = 1 + x + x^2 + x^3 + \cdots \] This is valid for \(|x| < 1\). In this problem, we want the Maclaurin polynomial of degree 4, which means the geometric series up to the \(x^4\) term.

Write the Polynomial of Degree 4

Using the geometric series expansion, the Maclaurin polynomial of degree 4 for \(f(x) = \frac{1}{1-x}\) is: \[ P_4(x) = 1 + x + x^2 + x^3 + x^4 \] where \(P_4(x)\) represents the polynomial including up to the \(x^4\) term.

Verify the Terms Match Degree

We confirm that the polynomial we have written has terms up to \(x^4\), which matches the requirement of the degree being 4, ensuring all relevant terms up are included.

Key concepts

Taylor Series

The Taylor series is a powerful mathematical tool used to express a function as an infinite sum of terms. These terms are derived from the values of a function's derivatives at a single point. In particular, the Taylor series centered at zero is called the Maclaurin series. This series can be written for any function that is infinitely differentiable at that point. Each term of the series involves higher-order derivatives of the function, divided by factorials to account for the number of times each derivative is repeated, multiplied by powers of the variable. For example, the term involving the second derivative will have an \(x^2\) term divided by \(2!\), and so on. This series provides a polynomial approximation that can be incredibly accurate near the point of expansion.

Geometric Series

The geometric series is a simplified series characterized by a constant ratio between consecutive terms. Often seen in its basic and recurring formula: \[ a + ar + ar^2 + ar^3 + \cdots \] where \(a\) is the initial term and \(r\) is the common ratio. When \(r\) is less than 1, the series converges, producing a finite sum. The formula for the sum of an infinite geometric series with \(|r| < 1|)\) is: \((1 - r)^{-1\). A classic example is the function \(f(x) = \frac{1}{1-x}\) which expands into \([1 + x + x^2 + x^3 + \cdots)\). This becomes the basis for finding the Maclaurin polynomial of a certain degree, by selecting terms up to that degree.

Polynomial Degree

In mathematics, the degree of a polynomial is defined as the highest power of the variable present. For instance, a polynomial like \(x^4 + 3x^3 + 7x^2 + x + 2\) is said to have a degree of 4. When constructing a Maclaurin polynomial of a specific degree, we simply truncate the higher-order terms that exceed the desired degree. This degree dictates how many terms we include from the series and how accurate the approximation of the original function might be within a local neighborhood of the expansion point. For example, the degree 4 Maclaurin polynomial of \(f(x) = \frac{1}{1-x}\) is \((1 + x + x^2 + x^3 + x^4)\).

Derivative

Derivatives are fundamental tools in calculus, representing the rate of change of a function with respect to a variable. When examining functions through Taylor and Maclaurin series, derivatives at a specific point determine each term in the expansion. For any function \(f(x)\), its first derivative \(f'(x)\) is the slope, second derivative \(f''(x)\) indicates concavity, and higher derivatives provide further insights into the function's behavior. During a series expansion, these derivatives evaluate how much the function changes as the variable shifts infinitesimally, forming each term's coefficient after adjusting for factorial growth. Thus, calculating precise derivatives is essential when forming polynomial approximations like the Maclaurin series.