Definition and Terminology
The number $e$ is defined as the limit of the sequence $a_n = \left(1+\dfrac{1}{n}\right)^{n}$ as $n\to\infty$:
$$ e \;:=\; \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} $$The symbol $:=$ emphasises that this is a definition: the number $e$ is introduced as the value to which the sequence converges.
Its decimal expansion begins $e \approx 2.71828\,18284\,59045\ldots$ The number $e$ is irrational (proved by Euler, 1744) and in fact transcendental (proved by Hermite, 1873). We state these facts without proof.
Numerical Exploration
Let us compute $a_n$ for several values of $n$:
| $n$ | $a_n = \left(1+\frac{1}{n}\right)^n$ |
|---|---|
| $1$ | $2.00000$ |
| $2$ | $2.25000$ |
| $5$ | $2.48832$ |
| $10$ | $2.59374$ |
| $100$ | $2.70481$ |
| $1\,000$ | $2.71692$ |
| $10\,000$ | $2.71815$ |
| $100\,000$ | $2.71827$ |
The terms increase steadily and approach $e\approx 2.71828$ from below. The convergence is monotone but slow: even at $n=1000$ the approximation agrees with $e$ only to the second decimal place.
Why the Limit Exists — the Monotone-Bounded Argument
To prove that the sequence $\{a_n\}$ converges we show two things:
- The sequence is strictly increasing ($a_{n+1}>a_n$ for all $n$).
- The sequence is bounded above ($a_n < 3$ for all $n$).