Approximate $\sqrt{e}$ by hand I have seen this question many times as an example of provoking creativity. I wonder how many ways there are to approximate $\sqrt{e}$ by hand as accurately as possible. 
The obvious way I can think of is to use Taylor expansion. 
Thanks 

Use the Power series to compute $x_n := \exp\left(-2^{-n}\right)$ for some $n \geqslant 1$ with high accuracy, and then compute
$$\sqrt{e} = \left(\frac{1}{x_n}\right)^{2^{n-1}}.$$
Using the negative exponent, and an exponent of smaller absolute gives you (much) faster convergence of the series, and the few operations of squaring and inverting don't lose much precision then. For e.g. $n = 3$, you get pretty good results for the first $10$ terms of the power series already.
