Problem 45 - Calculus: Early Transcendentals

Question

Find the Maclaurin series of \( f \) (by any method) and its radius of convergence. Graph \( f \) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \( f? \) \( f(x) = \cos (x^2) \)

Step-by-Step Solution

Answer

Maclaurin series: \(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{4n}\); Radius of convergence: Infinity. Taylor polynomials approximate better near \( x = 0 \).

Step 1: Understanding the Function and Maclaurin Series

We need to find the Maclaurin series of the function \( f(x) = \cos(x^2) \). A Maclaurin series is a special case of Taylor series centered at 0, which is given by \( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). First, let's recall that the Maclaurin series for \( \cos(x) \) is \( \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} \).

Step 2: Substitute and Adjust for \( f(x) = \cos(x^2) \)

Since our function is \( \cos(x^2) \), we substitute \( x^2 \) for \( x \) in the series for \( \cos(x) \). Thus, the series becomes \( \cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} (x^2)^{2n} = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{4n} \).

Step 3: Calculate Radius of Convergence

The radius of convergence for the standard cosine series \( \cos(x) \) is infinite. After substituting \( x^2 \) for \( x \), the behavior is still similar since the series involves even powers of \( x \). Thus, the radius of convergence, given the transformations, still extends over all real numbers. Therefore, the radius of convergence of \( \cos(x^2) \) is also infinite.

Step 4: Graphing the Function and Polynomials

Let's graph \( f(x) = \cos(x^2) \) along with its first few Taylor polynomials. Start with the constant function, \( P_0(x) = 1 \), and add successive terms from the series (\( P_1, P_2, \ldots \)). Observe how these polynomials approximate \( \cos(x^2) \) more closely as more terms are added.

Step 5: Observe and Analyze the Graph

You should notice that near \( x = 0 \), the Taylor polynomials increasingly approximate \( \cos(x^2) \) better as you include more terms. This is typical behavior for Maclaurin series where the function and its polynomial approximation overlap well near the center of expansion (in this case, \( x = 0 \)).

Key Concepts

Radius of Convergence

The radius of convergence represents how far a Maclaurin or Taylor series converges towards the function it represents across the x-axis. Specifically, it is the distance from the center of expansion (usually around zero for Maclaurin series) within which the series is guaranteed to converge.
In the case of the Maclaurin series for a function like \( f(x) = \cos(x^2) \), we're leveraging an even power series with terms like \( x^{4n} \). This means that the nature of convergence is preserved.

• Since \( \cos(x) \) has a convergence radius extending infinitely, substituting \( x^2 \) keeps the same extension for \( \cos(x^2) \).
• Thus, its series will also converge for all real numbers, giving it an infinite radius of convergence.

This infinite radius highlights that we can trust the approximation to closely model the function across the entire real line.

Taylor Polynomials

Taylor polynomials are essential tools for approximating functions. They are derived from the Taylor series, which represents a function by summing its derivatives at a single point.
In our context, we're examining Maclaurin polynomials — Taylor polynomials specifically centered at zero. Consider:

• The zero-order polynomial \( P_0(x) = 1 \), which is simply the constant term, approximates the function very roughly.
• As we add terms like \( x^{4n} \), our polynomial reflects more of the function’s behavior, providing better approximations particularly for small values of \( x \).
• Each additional term corresponds to an extra derivative, making the polynomial closely mimic \( \cos(x^2) \).

Thus, higher-order Taylor polynomials exhibit increasing similarity to the function across its radius of convergence.

Function Approximation

Function approximation refers to how closely a polynomial can represent a function over a specific interval. The goal is to use a simpler polynomial expression to mimic the more complex behavior of functions like \( \cos(x^2) \).

• By implementing Taylor polynomials, we are creating a sequence of approximations that ideally match \( f(x) \) near the center of the series expansion, \( x = 0 \).
• These approximations are incredibly useful in calculations where evaluating the original function directly is complicated or impossible.
• It’s especially effective near the expansion point, where the first few polynomials may give a sufficiently accurate representation.

This approach provides a practical way to handle functions in computational mathematics and engineering where precise calculations are needed.

Graphing Functions

Graphing functions like \( \cos(x^2) \) alongside their Taylor polynomials give a visual insight into how well these approximations perform.

• Start by plotting \( f(x) = \cos(x^2) \) and then overlay graphs of the Taylor polynomials, starting from the simplest \( P_0(x) = 1 \) and adding terms like \( x^4 \), \( x^8 \), etc.
• Close to \( x = 0 \), each additional term substantially reduces the approximation error.
• Observe that as you increase the degree of the polynomial, the approximation improves over a wider range around \( x = 0 \).

This visual comparison demonstrates the powerful utility of Taylor series expansions and their ability to approximate complex functions with simple polynomials. Understanding these graphical relations deepens the comprehension of how function approximation works in practice.