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A Taylor Series Expansion is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function at a specific point by using the values and derivatives of the function at that point.
A Taylor Series Expansion is used when an exact representation of a function is not possible or practical. It is particularly useful in situations where only a few terms of the series are needed for an accurate approximation.
The general formula for a Taylor Series Expansion is f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n, where f(x) is the function, a is the point of expansion, and n represents the number of terms used in the series.
The remainder term in a Taylor Series Expansion is used to measure the error or the difference between the actual function and the approximation using a finite number of terms. It becomes smaller as more terms are added to the series, leading to a more accurate approximation.
Taylor Series Expansion is commonly used in physics, engineering, and other scientific fields to approximate functions and solve differential equations. It is also used in statistics, economics, and other areas of mathematics to model complex relationships and make predictions.