Problem 12 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x^{2} \sin x$$

Step-by-Step Solution

Answer

The Taylor series for \(x^2 \sin x\) at \(x=0\) is \(x^3 - \frac{x^5}{3!} + \frac{x^7}{5!} - \cdots\).

Step 1: Recall the Taylor Series for Sine

The Taylor series for \( \sin x \) at \( x=0 \) is a well-known series given by:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]This series continues infinitely. Our goal is to use this series to find the Taylor series of \( x^2 \sin x \).

Step 2: Multiply the Series by \( x^2 \)

We take the series for \( \sin x \) and multiply each term by \( x^2 \):\[ \begin{align*}\sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \x^2 \sin x &= x^2(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots) \&= x^3 - \frac{x^5}{3!} + \frac{x^7}{5!} - \cdots\end{align*} \]

Step 3: Write the General Term for the Series

The general term in the series \( x^2 \sin x \) can be found by identifying the pattern from the multiplication:\[ x^2 \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+3}}{(2n+1)!} \]This formula represents the Taylor series for \( x^2 \sin x \) at \( x=0 \). Each term increases the exponent and factorial in a consistent manner.

Key Concepts

Power Series

A power series is a representation of a function as a sum of terms. Each term is composed of a variable raised to a power, multiplied by a coefficient. The general form of a power series centered at a point, usually denoted as \[ a_n (x - c)^n \]is key to understanding how many functions can be expressed. This expression consists of:

• the variable \((x - c)\), raised to the power of \(n\)
• \(c\), the center of the series, around which the function is expanded

The power series is crucial in approximating functions because, within a certain radius of convergence, it can closely replicate the behavior of more complex or non-polynomial functions. These series provide tools for evaluating functions that are difficult to express in a simpler algebraic form.
They are particularly useful for functions like exponentials, logarithms, and trigonometric functions. Being able to express the sine function as a power series allows for computations in situations where the exact value isn’t easily accessible.

Sine Function

The sine function, denoted as \( \sin x \), is a fundamental building block in trigonometry. It describes the y-coordinate of a point on the unit circle as it moves around.
For calculus and analysis, expressing \( \sin x \) as a Taylor series expansion is exceptionally important:

• It allows for approximation and solving problems involving this function where a direct solution might be complex or difficult
• The sine function's symmetry and periodicity characteristics make it a frequent subject of study and application, especially in physics and engineering
• When expanded as a Taylor series at \( x=0 \), \( \sin x \) becomes:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]This representation captures the oscillating nature of the sine function through alternating signs and increasing powers

This series is derived using derivative calculations for sine and demonstrates how using power series can make complex trigonometric calculations more manageable.

Series Expansion

Series expansion is the technique used to break down complex functions into an infinite sum of simpler terms, usually involving powers of \(x\). This method is particularly beneficial for performing operations such as differentiation and integration, which are far simpler in series form than with the original function.

• For a function expanded into a Taylor series at a point, the expansion takes the form:\[ f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \cdots\]
• In the context of \( x^2 \sin x \), the application of series expansion allows us to manipulate and compute the derivatives and function values in a systematic way
• Utilizing the expansion of \( \sin x \) and multiplying by \( x^2 \), we craft a new series that reflects the behavior of the entire expression: \[ x^2 \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+3}}{(2n+1)!}\]

This approach of series expansion not only simplifies complex operations but also equips learners with tools to handle analytical problems effectively across various domains of mathematics and physical sciences.