Problem 79 - Single Variable Calculus

Question

In Exercises \(79-84\), determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(P(x)\) is a polynomial function of degree \(n\), then the Maclaurin series for \(P\) is \(P\). 80\. Suppose that \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) for \(x\) in \((-R, R)\), where \(R>0\) and \(f\) is odd. Then \(a_{2 n}=0\) for \(n \geq 0\).

Step-by-Step Solution

Answer

The statement is true. For an odd function represented as a Maclaurin series, the coefficients of its even-degree terms, \(a_{2n}\), must be zero for all \(n \geq 0\). This is because the oddness property, \(f(-x) = -f(x)\), requires the series representation of \(f(-x)\) to be built with only odd terms and the even-degree terms in the series representation for \(f(x)\) must be zero.

Step 1: Identify the Problem Statement

The problem statement asks to verify whether the coefficients \(a_{2n}\) of an odd function's Maclaurin series are equal to zero for all \(n \geq 0\).

Step 2: Recall the Properties of Odd Functions

An odd function has the property \(f(-x) = -f(x)\) for all \(x\). The given function f(x) is odd, which, if represented with a Maclaurin series, must satisfy this property.

Step 3: Write out the Maclaurin Series for f(x)

We are given that the Maclaurin series for the odd function \(f(x)\) is: \[f(x) = \sum_{n=0}^{\infty} a_n x^n\]

Step 4: Check the Series for x and -x

Let's check the Maclaurin series for both x and -x: For \(x\): \[f(x) = \sum_{n=0}^{\infty} a_n x^n\] For \(-x\): \[f(-x) = \sum_{n=0}^{\infty} a_n (-x)^n\] For odd functions: \[f(-x) = -f(x)\]

Step 5: Prove the Assertion

Now let's analyze the series. We are given that \(f(x)\) is an odd function. So, \[-f(x) = \sum_{n=0}^{\infty} a_n (-x)^n\] Multiply the series in step 4 for \(f(x)\) by -1: \[-f(x) = -\sum_{n=0}^{\infty} a_n x^n\] Now, compare the two expressions for \(-f(x)\). Notice that the series representation of \(f(-x)\) is built from the odd terms in the series representation of \(-f(x)\). This implies that the even-degree terms in the series representation for \(f(x)\) must be zero in order for the oddness property to be preserved. Hence, \(a_{2n} = 0\) for all \(n \geq 0\). Therefore, the statement is true.

Key Concepts

Polynomial Function

Polynomial functions are mathematical expressions involving variables raised to whole number powers and multiplied by coefficients. For example, a simple polynomial is given by:

• \( P(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n \)

Here, each term of the polynomial is made up of a constant multiplier (the coefficient) and a variable (\(x\)) raised to a non-negative integer power. The highest power of the variable determines the degree of the polynomial, which in this example is \(n\).
Polynomials are important because they can represent a wide variety of functions and are used to model many real-world situations. They have straightforward operations such as addition, subtraction, and multiplication due to their simple algebraic structure. Polynomial functions are also very smooth and continuous, which makes them predictable and easy to work with in mathematics.

Odd Function

An odd function is a type of mathematical function characterized by a specific symmetry property. If \( f(x) \) is an odd function, then it satisfies the condition \( f(-x) = -f(x) \) for all values of \(x\).

• For example, the function \( f(x) = x^3 \) is odd because replacing \(x\) with \(-x\) yields \(f(-x) = (-x)^3 = -x^3 = -f(x)\).

Odd functions have a symmetric appearance around the origin of the coordinate plane. This means that the graph of an odd function will look the same if rotated 180 degrees around the origin. This rotational symmetry is a distinctive feature of odd functions and is essential in determining the coefficients in their series expansions, such as the Maclaurin series.

Coefficients

Coefficients are the numerical factors in front of terms involving variables in expressions, equations, or polynomials. They represent the proportion of each term in a polynomial expression and play a critical role in determining the properties and behavior of the function.

• In the Maclaurin series expansion, coefficients \(a_n\) indicate the contribution of \(x^n\) terms to the function \(f(x)\).
• For instance, in a series \(\sum_{n=0}^{\infty} a_n x^n\), each coefficient \(a_n\) corresponds to how much of the term \(x^n\) is included in \(f(x)\).

For odd functions, the coefficients for even powers of \(x\) are always zero. This ensures that the function remains odd, as demonstrated when using properties like \(f(-x) = -f(x)\). Understanding the role of coefficients helps in defining and manipulating polynomial functions and their series expansions.

Properties of Functions

Functions have various properties that determine their behavior and characteristics. These properties help in classifying functions and understanding their graphs and transformations.

• Symmetry: Functions can be symmetric about the y-axis, the origin, or neither. Odd functions, as mentioned, exhibit symmetry about the origin.
• Continuity: Functions are continuous if their graphs can be drawn without lifting a pencil from the paper. Polynomial functions, for example, are always continuous.
• Periodicity: Some functions repeat their values at regular intervals; these are called periodic functions. In our context, polynomial functions are not periodic, but they do have predictable patterns due to their structure.

Grasping the core properties of functions, like those mentioned, makes it easier to analyze complex mathematical problems. These properties form the basis for more advanced mathematical concepts and applications.