I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For example, 22/7 gets you "ok" close (it is 3.14285714), but it's still off by quite a lot. Another example is: 7920/2521 (as mentioned here). That gives you the result of 3.14161047. That's very good, but still not that close to pi.
EDIT: Just to lesson the challenge, how about accurate to the tenth decimal point ;) (if you can do 20, then go for it!).
If anyone tackles this, I'll thank them a lot!
The rational approximations $\frac{p}{q}$ to $\pi$ that you get from truncating the continued fraction get you about twice as many digits as you might expect, and are about as good as it is possible to get: they satisfy (and this is a general property of continued fractions)
$$\left| \pi - \frac{p}{q} \right| < \frac{1}{q^2}$$
The first one that is accurate to $10$ digits, including $3$, is
$$\frac{104348}{33215} = 3.141592653 \color{red}{0119 \dots}$$
and note that $q$ itself is only $5$ digits long.