What is the Taylor Series for ( f(x) e^x ) About ( x -1 )?

Introduction to Taylor Series Expansion:

Taylor series is a fundamental concept in mathematics, used to represent a function as an infinite sum of terms. Each term is derived from the function's derivatives at a specific point. This series is particularly useful in approximating functions near a point, which can simplify complex calculations and provide insight into the behavior of the function.

The general form of the Taylor series for a function ( f(x) ) around a point ( x a ) is given by:

Where ( f^{(n)}(a) ) represents the nth derivative of ( f(x) ) evaluated at ( a ).

Understanding the Exponential Function ( e^x )

The exponential function ( e^x ) is one of the most important and widely studied functions in calculus. It is defined as the function that is its own derivative:

(frac{d}{dx}e^x e^x)

This unique property makes ( e^x ) a key function in many areas of science and engineering, particularly in growth and decay problems.

Taylor Series Expansion of ( e^x ) Around ( x -1 )

To find the Taylor series of ( e^x ) about ( x -1 ), we need to calculate the derivatives of ( f(x) e^x ) at ( x -1 ) and then use the Taylor series formula.

Step 1: Calculate the First Few Derivatives of ( e^x )

( f(x) e^x ) ( f'(x) e^x ) ( f''(x) e^x ) ( f'''(x) e^x ) ...

As shown, all derivatives of ( e^x ) are equal to ( e^x ). Now, we evaluate these derivatives at ( x -1 ).

Step 2: Evaluate the Derivatives at ( x -1 )

( f(-1) e^{-1} ) ( f'(-1) e^{-1} ) ( f''(-1) e^{-1} ) ( f'''(-1) e^{-1} ) ...

Since all derivatives are equal to ( e^{-1} ), we can see that each term in the Taylor series will be ( e^{-1} ) multiplied by the corresponding factorial.

Constructing the Taylor Series

Using the Taylor series formula, we can now write the series expansion of ( e^x ) about ( x -1 ) as follows:

( e^x e^{-1} [1 (x 1) frac{(x 1)^2}{2!} frac{(x 1)^3}{3!} ... ] )

Let's break down this formula:

( e^{-1} ) is the constant term. ( (x 1) ) represents the first-order term, scaled by ( e^{-1} ). ( frac{(x 1)^2}{2!} ) is the second-order term, also scaled by ( e^{-1} ).

Benefits and Applications of Taylor Series Expansion

Taylor series expansions are incredibly useful in many areas:

Approximations: They allow us to approximate functions using polynomials, which are often easier to work with. Integration and Differentiation: Taylor series can simplify these operations by breaking down complex functions. Engineering and Physics: They are used in various models and simulation techniques where precise calculations are necessary.

Conclusion

The Taylor series expansion of ( e^x ) about ( x -1 ) is a powerful tool that leverages the unique properties of the exponential function. By understanding and implementing this series, we can better analyze and solve problems involving ( e^x ).