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Definition D1932
Standard natural real exponential function

The standard natural real exponential function is the D5482: Positive real function $$\mathbb{R} \to (0, \infty), \quad x \mapsto \lim_{N \to \infty} \sum_{n = 0}^N \frac{x^n}{n!}$$

The standard natural real exponential function is the D5482: Positive real function $$\mathbb{R} \to (0, \infty), \quad x \mapsto \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \frac{x^6}{6!} + \dots$$

The standard natural real exponential function is the D5482: Positive real function $$\mathbb{R} \to (0, \infty), \quad x \mapsto \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$
Children
 ▶ Napier's constant ▶ Softmax function
Results
 ▶ R4900 ▶ R4873 ▶ Napier's constant equals a limit of products with factors nearing one ▶ Standard approximating sequence for the natural exponential function ▶ Standard natural real exponential function maps zero to one
Conventions
 ▶ Convention 0 (Notation for standard natural basic real exponential function) The notation used for the D1932: Standard natural real exponential function is $x \mapsto \exp(x)$. Due to result R3621: Standard natural exponential function equals powers of Napier's constant, one may also use $x \mapsto e^x$.