Problem 71 - Thomas Calculus

Question

By multiplying the Taylor series for \(e^{x}\) and \(\sin x,\) find the terms through \(x^{5}\) of the Taylor series for \(e^{x} \sin x .\) This series is the imaginary part of the series for \begin{equation} e^{x} \cdot e^{i x}=e^{(1+i) x} \end{equation} Use this fact to check your answer. For what values of \(x\) should the series for \(e^{x}\) sin \(x\) converge?

Step-by-Step Solution

Answer

The series for \(e^{x} \sin x\) is \(x + \frac{5x^3}{6} + \frac{59x^5}{120}\), which converges for all \(x\).

Step 1: Write Taylor Series for \(e^{x}\)

The Taylor series for \(e^{x}\) is:\[ e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots \]

Step 2: Write Taylor Series for \(\sin x\)

The Taylor series for \(\sin x\) is:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]

Step 3: Multiply Both Series

Multiply the series obtained in Step 1 and Step 2 term by term and collect like terms up to \(x^5\):1. Constant Term: \(0\)2. \(x\) Term: \(x\)3. \(x^2\) Term: \(0\)4. \(x^3\) Term: \(x^3 - \frac{x^3}{6} = \frac{5x^3}{6}\)5. \(x^4\) Term: \(0\)6. \(x^5\) Term: \(\frac{x^5}{2} - \frac{x^5}{120} = \frac{59x^5}{120}\)Thus, \[ e^x \sin x \approx x + \frac{5x^3}{6} + \frac{59x^5}{120} \]

Step 4: Verify Using \(e^{(1+i)x}\)

The series \(e^{(1+i)x}\) is given by:\[ e^{(1+i)x} = e^{x} \times e^{ix} \]Since \(e^{ix} = \cos x + i \sin x\), we can focus on the imaginary part:\(e^{x}(i \sin x) = i( x + \frac{5x^3}{6} + \frac{59x^5}{120})\). This confirms that our multiplication was carried out correctly.

Step 5: Determine Convergence

Both series, \(e^{x}\) and \(\sin x\), converge for all real numbers \(x\). Therefore, the product \(e^{x} \sin x\) also converges for all real numbers \(x\).

Key Concepts

Exponential Function

The exponential function, denoted as \( e^x \), is a fundamental mathematical function that appears frequently in various areas of mathematics and the sciences. The function is defined as \( e^x = e^{ ext{power of } x} \), where \( e \) is a mathematical constant approximately equal to 2.71828. This function is particularly significant due to its unique property where the function is its own derivative. This means that the rate of change of \( e^x \) with respect to \( x \) is itself \( e^x \).

The Taylor series expansion for \( e^x \) around \( x = 0 \) is an infinite sum:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

This series converges for all real numbers \( x \), meaning the infinite polynomial is equal to \( e^x \) everywhere on the real line. Understanding this series is essential as it is the foundation for manipulating exponential functions in calculus and analysis.

Trigonometric Function

Trigonometric functions, such as sine and cosine, are central in mathematics, modeling periodic phenomena like sound waves and circular motion. Specifically, the sine function, denoted \( \sin x \), varies between -1 and 1, repeating every \( 2\pi \) radians. It can be expressed as an infinite series:

• \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)

The Taylor series for \( \sin x \) reflects its wave-like nature by alternating signs and involving only odd powers of \( x \).

This expansion is useful for approximating \( \sin x \) for small angles, where the series converges rapidly to give accurate results. Just like the exponential function, the convergence of the sine series also covers all real numbers \( x \), making it a versatile tool in both theoretical and applied math.

Series Multiplication

Series multiplication involves combining two infinite series by multiplying them term by term and collecting like terms. This process plays a crucial role in building new functions from known ones, especially when dealing with Taylor or power series expansions.

In our particular exercise, we have series for both \( e^x \) and \( \sin x \), and by multiplying these, we create a series for \( e^x \sin x \). Here's how you handle it:

• Multiply each term in the \( e^x \) series by each term in the \( \sin x \) series.
• Collect like terms based on the power of \( x \) to simplify.
• For terms up to \( x^5 \), carefully sum contributions from terms of the same power.

Using this method, you correctly construct the series to ensure accuracy up to the desired degree. This is particularly important in analytical studies where precise approximations are needed.

Convergence of Series

Convergence of a series refers to whether the sum approaches a specific value as more terms are added. For Taylor series, convergence determines if the series accurately represents the underlying function for most or all values of \( x \).

In our exercise, the convergence criterion is straightforward. The individual series for \( e^x \) and \( \sin x \) both converge for all real numbers. Thus, their product, \( e^x \sin x \), also converges for all \( x \). This fact is fundamental in mathematical analysis and ensures the reliability of using such series for practical applications across a wide array of \( x \) values.

• If a series converges, the approximation becomes highly accurate as more terms are added.
• For functions like \( e^x \) and \( \sin x \), their complete convergence means they can be used reliably across all real inputs.