Euler's Number as Limit of 1 + Reciprocal of n to nth Power

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Theorem

$\displaystyle \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = e$

where $e$ denotes Euler's number.


Proof 1

By definition of the real exponential function as the limit of a sequence:

$(1): \quad \exp x := \displaystyle \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$

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