Problem 12 - Thomas Calculus

Question

Find the Maclaurin series for the functions. \(x e^{x}\)

Step-by-Step Solution

Answer

The Maclaurin series for \( x e^x \) is \( x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + \cdots \).

Step 1: Understand the Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at 0. It is given by \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots \). We need to find the series for the function \( x e^x \).

Step 2: Recall the Exponential Function Expansion

The Maclaurin series expansion for \( e^x \) is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \). We will use this expansion to help find the series for \( x e^x \).

Step 3: Multiply the Series by x

To find the series for \( x e^x \), we multiply every term in the expansion of \( e^x \) by \( x \). Thus, we get \( x(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots) = x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + \cdots \).

Step 4: Write the Maclaurin Series Result

The Maclaurin series for \( x e^x \) is \( x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + \cdots \), which can be written in summation notation as \( \sum_{n=1}^{\infty} \frac{x^n}{(n-1)!} \).

Key Concepts

Taylor series

The Taylor series is an essential concept in calculus, allowing us to represent functions as infinite sums of terms. It is particularly useful for approximating functions that can be difficult to work with directly. At the core, a Taylor series for a function \( f(x) \) is an expansion around a certain point \( a \) and is expressed as:
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]
This series is particularly powerful because it allows us to approximate the function \( f(x) \) using only its derivatives at a single point \( a \). For many familiar functions, especially those common in mathematical and scientific calculations, their Taylor series can be found easily and used for analysis. In the case of the Maclaurin series, \( a = 0 \), simplifying the representation significantly.

exponential function

The exponential function \( e^x \) is a key mathematical function with numerous applications in science and engineering. Its Maclaurin series expansion is particularly straightforward and is given by:
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]
This expansion shows how the exponential function can be represented as an infinite sum of powers of \( x \), with coefficients involving factorials. The exponential function is unique because its derivative is also \( e^x \), leading to very predictable behavior when it is differentiated or integrated. This property makes it very useful when dealing with problems in calculus and differential equations.

series expansion

A series expansion expresses a function as an infinite sum of terms, providing a powerful tool for approximation and analysis. The Maclaurin series, as a special case of the Taylor series, is centered at 0, making it particularly simple and useful for functions defined at or near this point. For example, any function can be expressed by its derivatives at 0 if it is infinitely differentiable.

• Benefits of Series Expansion: Allows for easier numerical computations.
• Helps in understanding the behavior of functions near specific points.
• Crucial for solving differential equations analytically.

The series expansion of functions like \( e^x \) or trigonometric integrals often provide insights into their properties and are foundational in real-world applications.

multiplication of series

Multiplying series is a fundamental technique to derive new series, especially when dealing with functions like \( x e^x \). Given a known series like the expansion of \( e^x \), we can multiply the whole series by \( x \) to find the series of \( x e^x \):
\[x(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots) = x + x^2 + \frac{x^3}{2!} + \frac{x^4}{3!} + \cdots\]
This technique helps in expanding products of known series and is a crucial step in creating accurate approximations for composite functions. Multiplying series requires attentive distribution across each term, maintaining order and precision, which is vital for ensuring the result's accuracy in scientific computations. This approach is particularly helpful in enhancing expressions of functions where direct computation can be challenging.