e as the limit of (1 + 1/n)^n

Statement

Let $e = \exp(1)$, where $\exp$ is the exponential function. Then $$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e.$$ More generally, for each real number $x$, $$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x.$$

Remarks

Some texts take the limit as the definition of the number e. The sequence $(1 + 1/n)^n$ is increasing and bounded above by $3$, so the monotone convergence theorem gives the limit. The first values are $2$, $2.25$, about $2.370$ and about $2.441$. The limit is $e \approx 2.71828$. Jacob Bernoulli found this limit when he studied compound interest. Mathlib states the general form as Real.tendsto_one_add_div_pow_exp.

Explanations

L1 human

Put one unit of money in a bank at $100\%$ yearly interest, paid $n$ times a year. After one year you have $(1 + 1/n)^n$ units. As $n$ grows, this approaches $e$, and it never exceeds $e$.