91Ó°ÊÓ

The Taylor series is a way to represent functions as an infinite sum of terms, calculated from the values of their derivatives at a single point. The general form for the Taylor series of a function \( f(x) \) around \( x=0 \) (also known as the Maclaurin series) is given by: \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n} \].
Here, \( f^{(n)}(0) \) is the \( n \)-th derivative of \( f(x) \) evaluated at \( x=0 \), and \( n! \) denotes the factorial of \( n \).
This series is very useful because many functions that are difficult to deal with otherwise can be approximated by a simpler polynomial series.
In practice, we often use the first few terms of the Taylor series to approximate the function around a particular point.

Exponential functions are of the form \( e^x \), where \( e \) is the base of the natural logarithms, approximately equal to 2.71828.
These functions grow very quickly and have unique properties, such as the rate of growth being proportional to the value of the function.
The Taylor series for the exponential function \( e^x \) around \( x=0 \) can be written as: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \].
This series converges for all values of \( x \), which makes it a powerful tool for approximations.
In the context of the given problem, we are dealing with the function \( f(x) = e^{x} - e^{-x} \). This function combines two exponential functions, one positive and one negative.

A derivative represents the rate at which a function is changing at any given point.
For a function \( f(x) \), the first derivative \( f'(x) \) represents the slope of the function at point \( x \).
Higher-order derivatives (like the second derivative \( f''(x) \), third derivative \( f'''(x) \), and so on) provide more information about the curvature and the changing rates of the function.
In Taylor series, to find the coefficients of each term in the series, we compute multiple derivatives of the function up to the desired degree and evaluate them at \( x=0 \).
For the given function \( f(x) = e^x - e^{-x} \), here are the calculated derivatives: \[ f'(x) = e^x + e^{-x} \] \[ f''(x) = e^x - e^{-x} \] \[ f'''(x) = e^x + e^{-x} \] and so on.
When evaluating these derivatives at \( x=0 \), we notice a pattern where the odd derivatives are 2, and the even derivatives are 0. This pattern helps us construct the series efficiently.