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Taylor Series Definition ❓
The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point. The Series form of the Taylor series, known as the Taylor Series, approximates a function with a finite-degree Series.
Taylor Series Formula
The \( n \)th-degree Taylor Series for a function \( f(x) \) centered at a point \( a \) is given by:
\[P_n(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \frac{f'''(x_0)}{3!}(x-x_0)^3 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n\]
Where:
- \( f(x_0) \), \( f'(x_0) \), \( f''(x_0) \), \( f'''(x_0) \), etc., represent the values of the function and its derivatives evaluated at \( x = a \).
- \( n \) is the degree of the Series.
- \( f^{(n)}(x_0) \) denotes the \( n \)th derivative of \( f(x) \) evaluated at \( x = a \).
Usage
The Taylor series provides a method for approximating a function using a Series, which can be useful in various mathematical and computational contexts, such as numerical analysis, physics, and engineering. It allows the approximation of complex functions with simpler functions, facilitating analysis and computation.
Precision and Convergence
The accuracy of the Taylor Series approximation depends on the function and the degree of the Series used. As the degree of the Series increases, the approximation typically improves, especially in the vicinity of the expansion point \( x = a \).
However, it is important to note that the Taylor series only converges to the original function within a certain radius of convergence around the expansion point. Beyond this radius, the approximation may not be valid.
Example
Consider the function \( f(x) = e^x \) centered at \( a = 0 \) (Maclaurin series). The Maclaurin series expansion of \( e^x \) is:
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\]This series can be used to approximate \( e^x \) for different values of \( x \).
Conclusion
The Series form of the Taylor series provides a powerful tool for approximating functions with Series, enabling simplified analysis and computations in various mathematical and scientific domains.