Problem 15 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \cos \pi x$$

Step-by-Step Solution

Answer

The Taylor series for \(x\cos(\pi x)\) at \(x=0\) is \(x - \frac{\pi^2}{2}x^3 + \frac{\pi^4}{24}x^5 - \frac{\pi^6}{720}x^7 + \ldots\).

Step 1: Recall the Taylor series for cosine

The Taylor series for the cosine function centered at 0 is given by: \[ ext{cos}(x) = extstyleegin{align}egin{aligned} ext{1} - rac{x^2}{2!} + rac{x^4}{4!} - rac{x^6}{6!} + ext{...} = extstyleegin{align}egin{aligned} ext{ extstyleegin{align}egin{aligned} ext{ ext{}}} ext{1} + ext{i} ext{i}}}}}}}}dots \] This series will be used by substituting \(x\) with \(\pi x\).

Step 2: Substitute and simplify

Substitute \(\pi x\) into the Taylor series expansion for cosine: \[ ext{cos}( ext{πx}) = 1 - rac{( ext{πx})^2}{2!} + rac{( ext{πx})^4}{4!} - rac{( ext{πx})^6}{6!} + ext{...} \] This simplifies to: \[ 1 - rac{ ext{π}^2 x^2}{2!} + rac{ ext{π}^4 x^4}{4!} - rac{ ext{π}^6 x^6}{6!} + ext{...} \] Each term now involves \(x\) raised to an even power.

Step 3: Multiply by x to find the series of xcos(πx)

Multiply the function \(x\) by the series of \( ext{cos}( ext{πx})\): \[ x( ext{cos}( ext{πx})) = x\left(1 - \frac{\text{π}^2 x^2}{2!} + \frac{\text{π}^4 x^4}{4!} - \frac{\text{π}^6 x^6}{6!} + \text{...}\right) \] Distribute \(x\) through the series: \[ x - \frac{\text{π}^2 x^3}{2!} + \frac{\text{π}^4 x^5}{4!} - \frac{\text{π}^6 x^7}{6!} + \text{...} \] This is the series of \(x\cos(\text{πx})\).

Step 4: Write the final expression

The Taylor series for \(x\cos(\text{πx})\) at \(x=0\) is: \[ x - \frac{\text{π}^2 x^3}{2} + \frac{\text{π}^4 x^5}{24} - \frac{\text{π}^6 x^7}{720} + \text{...} \] Each term is obtained by multiplying \(x\) with the corresponding term of the cosine series expanded with \(\pi x\).

Key Concepts

Power Series

A power series is a way to represent a function as an infinite sum of terms involving powers of a variable. It is expressed in the form:

• Sum of \(a_n x^n\)
• Where \(a_n\) are coefficients
• \(x\) is a variable

The key thing about a power series is that it provides a tool to approximate functions. This is especially helpful when dealing with complex functions or when it's difficult to compute exact values.

In mathematics, a power series can converge, which means the sum becomes a finite value, or diverge, meaning it doesn't sum to a specific value. Convergence depends on the application and the values of \(x\).

Using power series allows us to understand and use functions in ways that can't be done using simple algebra. It's like having a magic formula that can simplify complex tasks. This concept is particularly crucial when dealing with the series expansion of functions like the cosine function.

Cosine Function

The cosine function, denoted as \( ext{cos}(x)\), is one of the basic trigonometric functions. It depicts the ratio of the adjacent side to the hypotenuse of a right-angled triangle. However, in calculus and more advanced mathematics, we often explore it beyond its geometric definition.

The Taylor series offers a way to express \(\text{cos}(x)\) as an infinite series centered around 0. This series is:

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

This expansion makes it easier to compute the cosine of an angle using a sum of polynomial terms. Each term alternates between positive and negative, and the power of \(x\) increases by even numbers.

This series is particularly useful when working with angles beyond the usual range or when using a computer or calculator. By substituting variables into this series, like substituting \(\pi x\) for the problem in question, it adjusts the calculation to suit various applications.

Series Expansion

Series expansion involves expressing a feature of a mathematical function as an infinite sum of terms based on powers of a variable. In the context of this problem, we dealt with a Taylor series, which is a type of series expansion.

A Taylor series gives a function's idea as an infinite sum around a certain point, often 0. This is called the function's expanded form and shows how the function behaves around this specific point.

For the cosine function, we modified the general Taylor series expression to accommodate \(\pi x\) and further adjusted for \(x\cos(\pi x)\).

• Start with the known cosine series
• Substitute with \(\pi x\)
• Multiply the series by \(x\) to achieve the desired function
• Each step transforms the series for specific applications

This systematic approach allows us to expand functions more suited to specific conditions while maintaining mathematical rigor. This series gives us a way to consider the nuances of functions through a simple, albeit infinite, polynomial representation.