Problem 25 - Calculus: One and Several Variables

Question

Expand \(g(x)\) as indicated. $$g(x)=x \sin x \quad \text { in powers of } x.$$

Step-by-Step Solution

Answer

The power series expansion of \(g(x) = x \sin x\) in powers of \(x\) is \(x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \frac{x^8}{7!} + \cdots\).

Step 1: Identify Known Power Series

The power series for \(\sin(x)\) is known and can be stated as \(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\). This power series will be used in the expansion.

Step 2: Multiply the Power Series with \(x\)

The given function is \(x \sin x\), which consists of \(x\) multiplied with \(\sin x\). Therefore, to get the expansion of the given function, multiply each term of the power series for \(\sin(x)\) by \(x\). This yields \(x (\sin x) = x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \frac{x^8}{7!} + \cdots\).

Key Concepts

Sin Function

The sine function is one of the fundamental trigonometric functions, commonly abbreviated as \( \sin(x) \). It is periodic, which means it repeats its values in regular intervals.
It is defined for all real numbers and has outputs ranging between -1 and 1. The sine function is particularly significant in fields like physics, engineering, and mathematics due to its representation of wave-like phenomena.

• The sine function is often used in the context of right-angled triangles, where it represents the ratio of the length of the side opposite the angle to the hypotenuse.
• In the unit circle, \(\sin(x)\) corresponds to the y-coordinate of a point on the circle.

Understanding \(\sin(x)\) and its behavior is crucial for various applications involving oscillating systems and waves.

Polynomial Series

A polynomial series is a sum of terms, each consisting of a variable raised to a power, often called the degree, and multiplied by a coefficient. Polynomial series are a way to represent functions as sums, which can be finite or infinite.

• Polynomial series serve as approximations for more complex functions.
• They allow one to express functions in a more simplified or accessible form.
• The degree of each term can vary, and coefficients often derive from factorial expressions in the case of power series expansions.

In the context of the sine function, representing \(\sin(x)\) using a power series simplifies the mathematical manipulations and calculations.
This series approximation becomes more accurate as more terms are included.

Mathematical Expansion

Mathematical expansion involves expressing a function as a series of terms that can be more easily manipulated or analyzed. A common type of expansion is the power series expansion, where functions are written as infinite sums of powers of a variable.

• These expansions are useful in calculus and mathematical analysis, providing insights into the behavior of functions around specific points.
• For trigonometric functions like \(\sin(x)\), expansions enable function evaluations, especially when dealing with small values of \(x\).
• They also play a crucial role in solving differential equations and in numerical analysis.

By expanding \(\sin(x)\) as a series, mathematicians can better predict and understand its characteristics under various transformations, such as multiplying by \(x\) in the original exercise.

Trigonometric Series

Trigonometric series involve sums of products of trigonometric functions, like sine and cosine, often multiplied by polynomial terms. These series extend the applicability of basic trigonometric functions to more complex mathematical problems.

• Trigonometric series can represent periodic functions, making them ideal for analyzing waveforms, sound signals, and other cyclical phenomena.
• These series are particularly useful in Fourier analysis, where functions are expressed as sums of sine and cosine waves.
• The power series expansion of trigonometric functions like \(\sin(x)\) helps in breaking down these functions into simpler components.

Such series are invaluable tools in various scientific and engineering disciplines, aiding in data approximation and signal processing. Understanding the power series representations of trigonometric functions is crucial for leveraging these methods effectively.