The Taylor series for arctan (1) is:
1 - 1/3 + 1/5 - 1/7 ....
Since tan 45° is 1 and 45° is equivalent to π/4 radians, multiplying the above expression by 4 will yield π itself.
(I suppose we could also look at 6 arccos (0.5), but the first is better, because the argument is simply 1).
The trouble is, just look at the denominators: 2n-1 for the nth term. That means it's an incredibly slow convergence as shown on these screenshots from a program running on an emulated Sinclair ZX Spectrum. (The terms are numbered from 0 in the program). In these images, we see the first, second and third decimal places having stabilized at their true values after 24, 626 and 2456 steps respectively. Imagine trying to do this by hand!
I love the way the most fundamental of irrational numbers can be arrived at by such a simple series of rational numbers, even if the method is glacially slow!
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