How to show $\frac{19}{7}How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$?
Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ 
However, how could this be shown in a testing environment where one does not have access to a calculator?
My only thought is to use the Taylor series for $e^x$ with $x=1$ to calculate $\displaystyle e\approx\sum \limits_{n=0}^{7}\frac{1}{n!}=\frac{685}{252}=2.7182...$ 
However, this method seems very time consuming and tedious, finding common denominators and performing long division. Does there exist a quicker, more elegant way?

This is a little easier than the OP's calculation $e\gt\sum_{n=1}^7{1\over n!}={685\over252}\gt{19\over7}$, though not by much:  We can show $e^{-1}\lt{7\over19}$ via the truncation of the alternating series
$$e^{-1}\lt1-1+{1\over2}-{1\over6}+{1\over24}-{1\over120}+{1\over720}={360-120+30-6+1\over720}={265\over720}={53\over144}$$
and the cross multiplication (with some of the steps retained to make things easy to check by eye)
$$53\cdot19=1060-53=1007\lt1008=980+28=144\cdot7$$
