What does the Taylor series represent?

The Taylor series represents a way to express a function as an infinite sum of terms calculated from the function's derivatives at a single point. This method captures the local behavior of the function around that point, specifically the function value and all its derivatives, which allows for approximation of complex functions using simpler polynomial forms.

When you compute the Taylor series, you start with the value of the function at a specific point and then add terms that are based on the derivatives of the function at that point, scaled by factorial denominators. This gives you a powerful tool for approximation, especially when dealing with functions that are difficult to analyze in their original form. By truncating the series after a certain number of terms, you can create a polynomial that approximates the function to a desired level of accuracy over a specific interval.Thus, the correct answer indicates the fundamental nature of the Taylor series and its connection to derivatives in forming an approximation of functions.