91Ó°ÊÓ

The sine function, denoted as \( \sin x \), is a basic trigonometric function. Its Maclaurin series is an infinite series that represents the function as a sum of its derivatives at zero. This series provides a polynomial approximation to \( \sin x \) for small values of \( x \), which is very useful in calculus. The Maclaurin series for \( \sin x \) is:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]In this exercise, we focused on the first two non-zero terms: \( x \) and \( -\frac{x^3}{6} \). This truncation allows us to work with a simpler expression that still captures the essence of \( \sin x \) for the purpose of multiplication with another series.

The exponential function, \( \exp x \), is another fundamental function in mathematics, especially prevalent in calculus and differential equations. Its Maclaurin series expansion provides another convenient polynomial approximation. The series for \( \exp x \) is elegant due to its simplicity:\[ \exp x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]It is characterized by the consistent coefficient pattern where each term is the product of \( x^n \) divided by \( n! \). For our purpose, we looked at the first four terms of the series. These are critical when multiplying by the series for \( \sin x \) since they directly contribute to terms up to \( x^3 \) in our combined series expansion.

Series multiplication is a method used to find the series representation of a product of two functions, given their respective series. In our case, we multiply the partial Maclaurin series for \( \sin x \) and \( \exp x \). This multiplication involves taking each term of \( \sin x \) and multiplying it by each term of \( \exp x \), focusing only on those combinations resulting in terms up to \( x^3 \). For example:

• \( x \times 1 = x \)
• \( x \times x = x^2 \)
• \( x \times \frac{x^2}{2!} = \frac{x^3}{2} \)
• \( -\frac{x^3}{6} \times 1 = -\frac{x^3}{6} \)

After performing these calculations, we sum the resulting terms. This results in the expression \( x + x^2 + \frac{1}{3}x^3 \), which represents the first few terms of the Maclaurin series for \( (\sin x)(\exp x) \). This technique greatly simplifies the process of integrating two series into a new function's series representation.