Definition:Exponential Function/Real/Power Series Expansion

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Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function.

The exponential function can be defined as a power series:

$\exp x := \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$

The number $\exp x$ is called the exponential of $x$.

Exponential Series

The power series expansion of the exponential function:

$\map \exp z = 1 + \dfrac z {1!} + \dfrac {z^2} {2!} + \dfrac {z^3} {3!} + \cdots + \dfrac {z^n} {n!} + \cdots$

is known as the exponential series.