Home page for accesible maths Math 101 Chapter 2: Functions of a real variable 2.18 Euler’s number e 2.20 Properties of the exponential.

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2.19 The exponential function $\exp x$ or $e^{x}$

The exponential function is fundamentally important in analysis and its applications. The definition that is given here is the most useful for computing the function and studying its properties.

For any real value of $x$ , we define the exponential of $x$ by the series

\exp(x)=1+x+{{x^{2}}\over{2!}}+{{x^{3}}\over{3!}}+{{x^{4}}\over{4!}}+\dots+{{x% ^{n}}\over{n!}}+\dots.

The series converges since $n!$ grows so rapidly. In MATH113 we shall discuss in detail why this series converges. For now, we simple use the formula as a convenient means of defining the function.