Problem 37 - Essential 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 \(f ?\) $$f(x)=\cos \left(x^{2}\right)$$

Step-by-Step Solution

Answer

The Maclaurin series for \( f(x) = \cos(x^2) \) is \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n}}{(2n)!} \) with infinite radius of convergence; more terms give better approximation.

Step 1: Understand the problem

We want to find the Maclaurin series for the function \( f(x) = \cos(x^2) \) and determine its radius of convergence. The Maclaurin series is a specific case of the Taylor series centered at 0.

Step 2: Recall the Maclaurin series of cosine

The Maclaurin series for \( \cos(x) \) is given by \( \cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \). We can use this fundamental series to find the series for \( \cos(x^2) \).

Step 3: Substitute \(x^2\) into the series

Substitute \( x^2 \) for \( x \) in the Maclaurin series of \( \cos(x) \). This gives \( \cos(x^2) = \sum_{n=0}^{\infty} (-1)^n \frac{(x^2)^{2n}}{(2n)!} = \sum_{n=0}^{\infty} (-1)^n \frac{x^{4n}}{(2n)!} \).

Step 4: Determine the radius of convergence

The radius of convergence of \( \cos(x) \) is infinite, meaning this series is valid for all \( x \). Since replacing \( x \) with \( x^2 \) doesn't affect the infinite radius of convergence, the series for \( \cos(x^2) \) also has an infinite radius of convergence.

Step 5: Graph the function and its polynomials

Graph \( f(x) = \cos(x^2) \) alongside its first few Taylor polynomials: the constant function \( 1 \), then \( 1 - \frac{x^4}{2} \), and \( 1 - \frac{x^4}{2} + \frac{x^8}{24} \). Observe how each successive polynomial becomes a closer approximation to \( f(x) \) near the origin.

Step 6: Notice the relationship

Observe that the more terms we add in the Taylor polynomials, the more closely the polynomials approximate \( \cos(x^2) \) over a larger interval around \( x = 0 \). This demonstrates the nature of Taylor series as local approximations of the original function.

Key Concepts

Taylor Series

Taylor series are incredibly useful in mathematics, providing a way to approximate complex functions with simpler polynomial expressions. A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. When that point is zero, it's called a Maclaurin series. For the function given in the exercise, \(f(x) = \cos(x^2)\), employing the Maclaurin series is a fitting approach as it allows us to express the function in terms of a power series centered at zero.

The general form of a Taylor series for a function \(f(x)\) around the point \(a\) is:

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

When centered at zero, the formula simplifies as the specific case of the Maclaurin series.

To find the series for \( \cos(x^2) \), one substitutes \( x^2 \) into the well-known series for \( \cos(x) \), providing a power series expansion that holds true across its entire domain. This approach not only simplifies computation but also illustrates how small modifications to variables can transform well-established series into new, useful tools.

Radius of Convergence

When dealing with power series, a critical aspect is the radius of convergence. It is the range within which the Taylor series will successfully approximate the function. In simpler terms, it's the set of values for which the series converges to the actual function.

For the cosine function, \(\cos(x)\), the radius of convergence is infinite, meaning it converges for all real numbers. This infinite radius implies that the power series is valid everywhere on the real number line. When we extend this knowledge to \( \cos(x^2) \), we see the same holds true, as substituting \( x \) with \( x^2 \) does not affect the convergence provided by the infinite radius.

• Thus, for \( \cos(x^2) \), we confidently state the radius of convergence remains infinite.

Understanding the radius of convergence is crucial, especially in applications where an approximation needs to be valid over a specific domain. It tells us how far from the center point of the series we can expect the approximation to be accurate and reliable.

Graphing Taylor Polynomials

Visualizing Taylor polynomials allows us to see firsthand how they serve as approximations to their corresponding functions. When graphed together, as in the exercise with \(f(x) = \cos(x^2)\) and its Taylor polynomials, we gain insight into how these polynomials work.

Starting with the simplest form, the constant term, the polynomial begins as a flat line. Adding more terms increases its complexity:

• \(1 - \frac{x^4}{2}\) forms a parabola, improving the approximation around \(x=0\).
• \(1 - \frac{x^4}{2} + \frac{x^8}{24}\) adds more curvature, extending the accuracy of the approximation further away from zero.

By continuing to add terms, the Taylor polynomial increasingly resembles the original function. This refinement is notable near the center but less effective far from it.

This process illustrates how Taylor series reveal the "local" nature of functions around specific points, and why they are crucial in computations involving approximations, especially when only a few terms can be used efficiently. By graphing, students can visually connect the purely mathematical procedure of creating series with the tangible outcome of function approximation.