Problem 19 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$\cos ^{2} x\left(\operatorname{Hint} : \cos ^{2} x=(1+\cos 2 x) / 2 .\right)$$

Step-by-Step Solution

Answer

The Taylor series for \( \cos^2 x \) at \( x = 0 \) is \( 1 - x^2 + \frac{x^4}{12} - \cdots \).

Step 1: Use the Hint to Rewrite Function

The hint suggests using the identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \). This identity helps in simplifying the function as it separates \( \cos^2 x \) into terms that are easier to work with for series expansion.

Step 2: Expand \( \cos 2x \) using Taylor Series

The cosine function has the Taylor series expansion \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \). Applying this to \( \cos 2x \), we replace \( x \) with \( 2x \):\[ \cos(2x) = 1 - \frac{(2x)^2}{2} + \frac{(2x)^4}{24} - \cdots \] Simplifying, we get:\[ \cos(2x) = 1 - 2x^2 + \frac{(2x)^4}{24} - \cdots \]

Step 3: Divide the Expanded Series by 2

The identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \) requires a division by 2 of both terms in the expansion. So,\[\frac{1 + \cos 2x}{2} = \frac{1}{2} + \frac{1 - 2x^2 + \frac{(2x)^4}{24} - \cdots}{2}\]Simplifying, we get:\[\cos^2 x = \frac{1}{2} + \frac{1}{2} - x^2 + \frac{x^4}{12} - \cdots\]

Step 4: Simplify the Resulting Series

Combine the terms to simplify the series:\[\cos^2 x = 1 - x^2 + \frac{x^4}{12} - \cdots\]

Step 5: Write the Final Taylor Series

The Taylor series for \( \cos^2 x \) about \( x = 0 \) is:\[\cos^2 x = 1 - x^2 + \frac{x^4}{12} - \cdots\]This is the final series derived using power series operations.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the basic trigonometric functions that appears frequently in mathematics, particularly in calculus and geometry. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. However, in the context of calculus and series expansions, we often explore its infinite series representation.For small angles, the Taylor series of the cosine function centered around \( x = 0 \) (also known as the Maclaurin series) provides an effective way to approximate the value of \( \cos x \). It is given by:

• \( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \)

This series highlights how the cosine function can be expressed as an infinite sum of powers of \( x \). Each term consists of an alternating sign and involves even powers of \( x \). This alternating pattern is characteristic of the cosine series, contributing to its convergence properties when \( x \) is close to zero.

Power Series

A power series is an infinite series of the form:

• \( a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots \)

where \( a_0, a_1, a_2, \ldots \) are coefficients, and \( x \) represents the variable. Power series are crucial because they allow us to express complex functions as infinite polynomials, making them easier to manipulate and study.When functions such as trigonometric functions, exponentials, and logarithms do not have simple polynomial forms, we can use power series to represent these functions in a locally convergent form around a particular point, often referred to as the center of the series or the point of expansion. The Taylor series and Maclaurin series are specific types of power series expansions.Power series offer several advantages:

• Facilitating numerical calculations by utilizing only a few terms for approximation.
• Providing insight into the behavior of functions near a specific value (usually zero for Maclaurin series).
• Allowing for operations like differentiation and integration term by term.

Series Expansion

A series expansion involves expressing a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This technique helps translate complex or transcendental functions into simple mathematical expressions that are easier to work with, especially in calculus.The process of a series expansion can be understood through the Taylor series, where a function \( f(x) \) is expanded around a point \( x = a \) as follows:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \)

When \( a = 0 \), it becomes a Maclaurin series. Series expansion is particularly useful:

• For finding polynomial approximations of functions.
• In solving differential equations and complex integrals that are otherwise difficult to handle analytically.
• For understanding functions' behavior, especially near the point of expansion.

In the example of \( \cos^2 x \), decomposing the function and utilizing the series expansion of \( \cos 2x \) allowed us to derive a simplified series form for the entire expression, making further mathematical operations more straightforward.