Euler's Number as Limit of 1 + Reciprocal of n to nth Power
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Theorem
- $\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = e$
where $e$ denotes Euler's number.
Proof 1
By definition of the real exponential function as the limit of a sequence:
- $(1): \quad \exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
By definition of Euler's number:
- $e = e^1 = \exp 1$
The result follows by setting $x = 1$ in $(1)$.
$\blacksquare$
Proof 2
Euler's Number as Limit of 1 + Reciprocal of n to nth Power/Proof 2