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The exponential function, denoted as \( e^y \), is a fundamental mathematical function that can be expanded into a power series. This is particularly helpful in calculus as it allows us to express functions in a form that is easier to manipulate mathematically. The exponential function has the remarkable property of being equal to its own derivative, and it can be represented as:

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

Here, each term involves a power of \( y \) divided by the respective factorial. This series is especially useful when \( y \) is a function of \( x \), such as \( \sin x \) in our exercise. By substituting \( \sin x \) into the power series, we are able to tackle complex expressions like \( e^{\sin x} \) by breaking them down into simpler polynomial terms. This method simplifies calculations involving exponential functions considerably, turning them into a series of addition operations.

A Taylor series is a powerful mathematical tool used to approximate functions by infinite sums of their derivatives evaluated at a point. It provides a polynomial approximation of a function around a particular point, commonly chosen to be zero, known as a Maclaurin series. The general form of a Taylor series expansion for a function \( f(x) \) around zero is:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \)

When applied to the sine function, this becomes:

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

This series is an infinite sum, and in practice, we calculate only a few terms for an approximation. By using the Taylor series of \( \sin x \), we can substitute it into more complex functions, as done in our example where we find the expansion of \( e^{\sin x} \). This method allows us to express \( e^{\sin x} \) explicitly in terms of powers of \( x \), making it much easier to understand and compute.

The sine function, \( \sin x \), is one of the fundamental trigonometric functions that appears frequently in mathematics, particularly in calculus and physics. The Taylor series expansion of \( \sin x \) gives:

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

This series is significant because it transforms the periodic behavior of the sine function into an infinite polynomial form which can be used for various mathematical procedures like differentiation and integration. In the context of power series expansion, the sine function's series allows us to take complex compositions, like \( e^{\sin x} \), and break them into polynomial forms. Substituting this series into \( e^{\sin x} \) demonstrates how algebraic techniques can simplify trigonometric expressions, making them more accessible for further analysis or computational purposes.