values of ${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{i=0}^n\frac{1}{i!}}$ for

The following table gives the numerator and denominator of

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

as well as the decimal expansion to 20 places.

As the table shows, small values of $n$, are sufficient to give a good approximation of the natural log base $e$, which at $n=21$ is at 20 places undistinguishable from $e$.