calculus - How to prove $limlimits_{n to infty} (1+frac1n)^n = e$?

How to prove the following limit?

$$\lim_{n \to \infty} (1+1/n)^n = e$$
I can only observe that the limit should be a very large number!

Thanks.

Answer

Just apply the binomial theorem then move things around a bit: $$\left(1+\tfrac{1}{n}\right)^n = \sum_{k=0}^n \frac{{n \choose k}}{n^k} = \sum_{k=0}^n \frac{1}{k!} \frac{n!/(n-k)!}{n^k}.$$

Now if you could just get rid of that $\frac{n!/(n-k)!}{n^k}$ term... (using the idea of $n \to \infty$ and $k$ is usually small in comparison)

edit Jonas makes a good point, I was implicity assuming the definition $e = \sum_{k = 0}^\infty \frac{1}{k!}$