Here's an illustration of an approach to pi pointed out by Kevin Brown. Define f(n) as the nearest greater or equal multiple of n-1, then of n-2, etc (yielding OEIS sequence 2491). Then, inverting a result found by Duane Broline and Daniel Loeb, pi = n2 / f(n).
But as you can see from the comment, the series converges very slowly!
main = print $ e**2 / f e e 1
e = 900000 -- yields 3.1416003...
f n 1 _ = n
f n k l = f n' (k - 1) $ n' / k
n' = head $ dropWhile (< n) $ map (* k) [l..]