Maclaurin Series for ex sin x

Find the first three non-zero terms in the Maclaurin series for $e^x \sin x$.

The Maclaurin series is a specific form of Taylor series centered at zero. It provides a way to represent functions as infinite sums of terms calculated from the values of a function’s derivatives at a single point. In this problem, we are tasked to find the first three non-zero terms of the Maclaurin series for the product of the exponential function $e^x$ and the sine function $\sin x$. This involves understanding how to combine the series of two different functions and extracting terms of interest.

To approach this, it is critical to remember the basic Maclaurin series for both $e^x$ and $\sin x$. The series for $e^x$ is the sum of $\frac{x^n}{n!}$ for $n \geq 0$, and for $\sin x$, it alternates between positive and negative powers of odd exponents of $x$ over their respective factorials. When tasked to find terms of a product of series, one strategy is to use substitution and multiplication of two infinite series and then focus on collecting terms that match the target degree of accuracy, usually established by the first non-zero coefficients required by the problem.

The importance of such series in mathematical analysis cannot be overstated. Taylor and Maclaurin series not only allow us to approximate functions but also to analyze their behavior in a neighborhood around the point of expansion. They play crucial roles in numerical analysis, differential equations, and even in practical applications such as signal processing. Gaining proficiency with these series requires understanding differentiation and power series manipulation, as well as practice in recognizing patterns within infinite sequences. This particular problem challenges you to blend your understanding of these concepts, enhancing your skills in power series manipulations and their applications.