Wikipedia has a great article on methods for calculating pi with arbitrary precision, using for example Machin's infinite series expansion:
$\frac{\pi}{ 4} = 4$ arccot $5 - $arccot $ 239 $
where
arccot $x = \frac{1}{x} - \frac{1}{3 x^3} + \frac{1}{5 x^5} - \frac{1}{7 x^7} + \dots$
The only problem is that they never mention how to prove how many digits you have accurately computed pi to. How is this done? In general, how do you prove this for any series that converges?
Edit: I used pi with an alternating series as an example, but I'm more interested in general techniques for any convergent series.