Problem 52 - Thomas Calculus

Question

Taylor series for even functions and odd functions (Continuation of Section \(10.7,\) Exercise \(59 .\) ) Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) converges for all \(x\) in an open interval \((-R, R) .\) Show that \begin{equation} \begin{array}{l}{\text { a. If } f \text { is even, then } a_{1}=a_{3}=a_{5}=\dots=0, \text { i.e., the Taylor }} \\ {\text { series for } f \text { at } x=0 \text { contains only even powers of } x .} \\ {\text { b. If } f \text { is odd, then } a_{0}=a_{4}=a_{4}=\cdots=0, \text { i.e., the Taylor }} \\ {\text { series for } f \text { at } x=0 \text { contains only odd powers of } x .}\end{array} \end{equation}

Step-by-Step Solution

Answer

For even functions, the Taylor series includes only even powers; for odd functions, only odd powers.

Step 1: Define Even and Odd Functions

A function \( f(x) \) is called even if \( f(-x) = f(x) \). Conversely, a function is called odd if \( f(-x) = -f(x) \). We will use these definitions to show which terms in the Taylor series must be zero.

Step 2: Write the Taylor Series

The Taylor series for a function \( f(x) \) centered at \( x = 0 \) is given by \( f(x) = \sum_{n=0}^{fty} a_n x^n \). This expresses \( f(x) \) as an infinite sum of terms of the form \( a_n x^n \).

Step 3: Analyze the Series for an Even Function

If \( f(x) \) is even, then \( f(-x) = f(x) \), leading to \( \sum_{n=0}^{fty} a_n (-x)^n = \sum_{n=0}^{fty} a_n x^n \). This implies that the series terms must be equal for coefficients of like powers of \( x \), so terms where \( n \) is odd must cancel, i.e., \( a_1 = a_3 = a_5 = \dots = 0 \).

Step 4: Analyze the Series for an Odd Function

If \( f(x) \) is odd, then \( f(-x) = -f(x) \), leading to \( \sum_{n=0}^{fty} a_n (-x)^n = -\sum_{n=0}^{fty} a_n x^n \). All terms with even powers of \( x \) must cancel out, which means \( a_0 = a_2 = a_4 = \cdots = 0 \).

Step 5: Conclude on Taylor Series Structure

Thus, for an even function, the Taylor series at \( x=0 \) contains only even powers of \( x \), and for an odd function, it contains only odd powers. This conclusion is consistent with the properties of even and odd functions, as shown in the analysis of the series.

Key Concepts

Even Functions

An even function is one that satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. Trying this with simple functions, think about how the graph of \( x^2 \) looks—it's a perfect mirror across the y-axis.

In the context of a power series representation like the Taylor series, when we talk about an even function, only the even power terms in the series have coefficients that are not zero. This means the series will look something like:

• \( f(x) = a_0 + a_2 x^2 + a_4 x^4 + \ldots \)

For instance, the function \( \ cos(x) \) has a Taylor series expansion with only even powers. Understanding this helps us simplify calculations for such functions and predict their behavior.

Odd Functions

An odd function meets the requirement \( f(-x) = -f(x) \). This tells us it is symmetric about the origin; if you flip the graph both over the x-axis and then the y-axis, it stays the same. Functions like \( x^3 \) clearly show this property on their graphs.

In terms of their power series representation, only odd powers remain with non-zero coefficients. That means for an odd function, the Taylor series expansion only consists of odd powers:

• \( f(x) = a_1 x + a_3 x^3 + a_5 x^5 + \ldots \)

Consider the function \( \ sin(x) \) which is odd, and observe its Taylor series expansion—it contains only odd powers. This distinction is crucial when working with functions in calculus, allowing us to streamline our problem-solving approach by focusing only on the relevant terms.

Power Series

A power series is like an infinite polynomial, expressed in the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) represents the coefficients and \( x^n \) represents the variable raised to the nth power. Think of it as adding up terms, where the power of x keeps increasing.

These series are incredibly important in mathematics because they allow for functions to be represented and approximated in ways that are often easier to work with, particularly around the point of expansion (commonly x=0). They serve as the foundation for more complex mathematical concepts, particularly in calculus and analytic geometry.

• Power series can converge to give a finite value.
• They are used for approximating functions that are otherwise difficult to solve.
• The concept is fundamental in solving differential equations and in developing numerical methods.

Convergence

Convergence refers to the behavior of a series as its terms approach a single value as more terms are added. For a power series, convergence is essential because it tells us when the series accurately represents the function. Without convergence, the series might not provide a valid value for any computations.

This concept is defined over an open interval \((-R, R)\) where \( R \) is the radius of convergence, a point beyond which the series does not converge. You could think of it as the distance you can go from the center point (like x=0 in many Taylor series) and still have a valid series.

• The series converges absolutely if the sum is finite.
• Within the radius of convergence, the series becomes a reliable tool for function evaluation and manipulation.
• The concept of convergence is critical in ensuring that mathematical results are consistent and valid.

By understanding convergence, we can effectively apply power series to real-world problems, ensuring precision and accuracy in our solutions.