TITLE: Evaluating the precision in the calculation of $\mathrm{e}$
QUESTION [6 upvotes]: I'm calculating $\mathrm{e}$ using a computer like this:
$$
\mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!}
$$
I'm storing it as a rational number. 
I was wondering, if I write down my rational number as a decimal number, could I determine, 
how many digits after the decimal point are correct for a given value of $n$?

REPLY [10 votes]: As yet another possibility, if you calculate
$$\frac1e = \sum_{i=0}^\infty \frac{(-1)^i}{i!} $$
then you have an alternating series, so the true value of $1/e$ is strictly between any two successive partial sums, which you can then invert and represent in decimal. Any digits they agree on are certain.