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Source and Methodology
All calculations are based on the mathematical principles of Taylor series as detailed in this Wolfram Alpha resource. All computations strictly adhere to the described formulas.
The Formula Explained
The Taylor series of a function f at a point a is given by:
\( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \)
Glossary of Variables
- Function (f(x)): The mathematical function to expand.
- Point (a): The point at which the function is expanded.
- Number of Terms (n): The number of terms in the Taylor series expansion.
How It Works: A Step-by-Step Example
Consider the function \( f(x) = e^x \). To find the Taylor series expansion at \( a = 0 \) with 3 terms:
- Calculate derivatives: \( f^{(0)}(0) = 1 \), \( f^{(1)}(0) = 1 \), \( f^{(2)}(0) = 1 \)
- Apply the formula: \( T_3(x) = 1 + x + \frac{x^2}{2!} \)
Frequently Asked Questions (FAQ)
What is a Taylor series?
A Taylor series is an infinite sum of terms calculated from the values of the function's derivatives at a single point.
How many terms do I need?
The number of terms needed depends on the desired accuracy and the behavior of the function.
Can all functions be expanded into a Taylor series?
Not all functions can be expanded using Taylor series, especially if they are not differentiable at the point of expansion.
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is a Taylor series expanded at \( a = 0 \).
How do I choose the point of expansion?
The point should ideally be where the function is smooth and differentiable to ensure convergence.