Well, note that all your partial sums are rational numbers; you can store them exactly by storing the numerator and denominator exactly. And the numerator and denominator can be stored exactly as just long strings of digits (presumably, for speed reasons, in a base 2^k system, where your word size is k bits, so that they’re stored as just long strings of words).
Now, as for how to get an approximation to a desired level of precision, the useful series generally come with known asymptotically vanishing error bounds: rules that tell you an upper bound for your distance from the actual limit after you’ve computed the Nth partial sum, with the upper bound approaching 0 as N approaches infinity. Keep going till the error bound is small enough, and you have enough precision. Alternatively (and essentially equivalently), get two series that both converge to pi (or whatever value it is you’re calculating); one approaching it from below (always) and one approaching it from above (always). As you calculate the two partial sums, they give you tighter and tighter bounds on the value, and you can stop when you achieve the amount of precision you want. [This is what Cecil was getting at with using inscribed and circumscribed polygons of many sides to calculate the circumference of a circle].
Now, I’m afraid I can’t recall off the top of my head a good error bound for the Leibniz series but I’m pretty sure you could derive one using Taylor’s theorem (at any rate, as Cecil remarked, it converges extremely slowly). I can also tell you that, while the Leibniz series neither stays below nor above pi, the series 1/1^2 + 1/2^2 + 1/3^2 + … converges to pi^2/6 from below, from which you can extract an approximation which converges to pi from below. Similarly, there’s also Wallis’s formula: the infinite product (2^2)/(2^2 - 1) * (4^2)/(4^2-1) * (6^2)/(6^2-1) * (8^2)/(8^2-1) … converges to pi/2 from below. So using either of those two and some formula which converges to pi from above, you could (slowly) calculate pi to any desired precision. Also, another way to get series which converge to pi from below or from above is to use lower or upper Riemann sums to calculate the integral of sqrt(1-x^2) from x = 0 to x = 1, which is equal to pi/4. (Thus, from this, if you know how to compute square roots to arbitrary precision, you get an easy method for computing pi to arbitrary precision).
But all of those are fairly slow convergers. Real pi hunters often use a variation on Leibniz’s formula called Machin’s formula, which converges rather faster. Or they use various other algorithm’s (e.g., the Gauss-Legendre algorithm, or Borwein’s algorithm). I’m no numerical analyst, so I can’t tell you much about the convergence rates, but the point is, they’re out there, useful asymptotically vanishing error bounds are known, so you can eventually tell with any of these methods that you’ve reached the desired level of precision, even if it’s not something as simple as “Each step gives you precisely one more digit”. And like I said, it’s easy enough to store numbers exactly enough to do the calculations: if the calculations only involve rationals, you can store all those perfectly; similarly for if, say, the calculations only involve sums of K-th roots of rationals, you could store those all perfectly and do arithmetic with them, with a properly designed encoding system. If your calculations are more hairy than would allow for any of that, you can do interval arithmetic (store perfect rational lower and upper bounds on all your numbers, treating them as fuzzy intervals rather than precise values) or, essentially equivalently, you can do some computation to figure out what level of precision you need to work with in your calculations to get what level of precision out. Like I said, I unfortunately don’t know all the formulas telling you exactly how much you have to do to get out how much precision, but people who do this sort of thing do know such formulas.
[Since I’m no expert in this area, anyone who is a numerical analyst, or has any other reason to want to correct me, feel free to go ahead and do so on any point. But I think I’ve represented the situation quite accurately.]
You know, so it doesn’t optimize the fork() out of the loop.