[hinkley]

Oh, I mean they're both wrong in the 10th position even though they agree. But it is curious that they diverge after that.

It's probably more a problem of the algorithm not calculating significant figures and truncating the answer.

[kmill]

The algorithm is computing an alternating series, with the largest n being 5x10^7. The error in an alternating series (when the sequence is strictly decreasing in absolute value) is the absolute value of the next term. This is 4/(2*(5x10^7 + 1) + 1), which is a bit less than 4x10^-8, so we would only expect 7 digits to be correct. In fact, it is the 8th digit which goes awry (not the 10th): it should be a 5 rather than a 7.

This particular series, which comes from a series for the arctan of 1, has very slow convergence. An easily better one is Machin's formula, which converges faster than a geometric series (that is, the number of terms needed is at most linear in the number of wanted digits).

[e12e]

However, according to math.pi in python3.3 the second number (with higher number of iterations) is closer to pi than the lower iteration number.

So is it possible that the algorithm doesn't converge to correct digits for so "few" iterations?