The Taylor Series Expansion of the Exponential Function (e^x): A Comprehensive Proof

Understanding the Taylor series expansion of the exponential function (e^x) is fundamental in calculus and has numerous applications in various fields such as physics, engineering, and computer science. This article will delve into the derivation and significance of this series expansion, providing a detailed proof based on the Taylor series expansion formula around (x 0).

Introduction to Taylor Series

The Taylor series is a powerful tool in approximating functions, especially around a point (a). The general form of the Taylor series for a function (f(x)) around (x a) is given by:

[f(x) f(a) f'(a)(x-a) frac{f''(a)}{2!}(x-a)^2 frac{f'''(a)}{3!}(x-a)^3 cdots frac{f^{(n)}(a)}{n!}(x-a)^n cdots]

When (a 0), the series is referred to as the Maclaurin series. For the exponential function (f(x) e^x), we will investigate its Maclaurin series expansion.

The Derivatives of (e^x)

A remarkable property of the exponential function (e^x) is that all its derivatives are equal to the function itself. That is, for (n 0, 1, 2, 3, ldots):

[f^{(n)}(x) e^x]

Evaluating these derivatives at (x 0):

[f(0) e^0 1, quad f'(0) e^0 1, quad f''(0) e^0 1, quad f'''(0) e^0 1, quad ldots]

Plugging these values into the Maclaurin series formula:

[e^x 1 x frac{x^2}{2!} frac{x^3}{3!} frac{x^4}{4!} cdots]

This can be written more succinctly as:

[e^x sum_{n0}^{infty} frac{x^n}{n!}]

Proof of Convergence

The series (sum_{n0}^{infty} frac{x^n}{n!}) converges for all real values of (x). This property is crucial in ensuring that the series approximates the function (e^x) accurately across the entire real line.

Conclusion

The Taylor series expansion of the exponential function (e^x) is a remarkable mathematical concept with wide-ranging applications. Understanding its derivation and properties provides insight into the nature of exponential growth and decay, making it an essential tool in advanced mathematics and applied sciences.

For further reading, you may refer to:

Exponential Function - Wikipedia Radius of Convergence - Wikipedia Taylor series - Wikipedia

Thanks for your interest in this topic!