values of ${(1+1/n)}^n$ for

The following table gives the numerator and denominator of ${\left(1+\frac{1}{n}\right)}^n$ as well as the decimal expansion to 20 places.

With a large enough value of $n$, this formula approximates the natural log base $e$. For example, with $n$ set to ten million, we get 2.7182816925449662712, which is 0.000000135914078964161737245574 short of 2.7182818284590452354 (this calculation took almost four minutes with Mathematica 4.2). It is less computationally intensive to use

$$\sum _{i=0}^n\frac{1}{i!},$$

which with $n$ set to 100 gives in less than a second a result to 20 places that is indistinguishable from $e$.