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The classical series $e = \lim_{n \to \infty} X_n$ where $X_n = \sum_{k=0}^n 1/k!$ is incredibly efficient. But is it known to be the most efficient series in terms of denominators for using fractions in the sum? In other words, is it known whether there is another series of fractions $e = \lim_{n \to \infty} Y_n$ where $Y_n = \sum_{k=0}^n a_k/b_k$ ($b_k > 0$, $a_k,b_k$ integers) where $\lim_{k \to \infty} b_k/k! \leq 1$ and $\lim_{n \to \infty} |(Y_n - e)/(X_n -e)| < 1$?

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The classical formula $X_n = \sum_{k=0}^{k=n} \frac{1}{k!}$ was used by A. Yee, in 2010, to calculate the first 500 billion digits of e. Thus I guess using $X_n$ is still the state of art method to calculate $e$.

I have a vague reason for $X_n$ being a really good choice to calculate $e$. Observe the following , $\frac{1}{10!} = 2.7557319e^{-7}$, $\quad$ $\frac{1}{11!} = 2.5052108e^{-8}$, $\quad$$\frac{1}{12!} = 2.0876756e^{-9}$

As $k$ increases, the largest decimal value which $1/k!$ can effect increases. Therefore, one can calculate the first $l$ decimal places accurately by computing $X_n$ for finite $n$. Also, $n$ scales linearly with respect to $l$

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    $\begingroup$ That site was updated three days ago: 1 trillion digits are now available! $\endgroup$ – TonyK Jun 20 '14 at 11:46

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