>>13940877You get a decreasing alternating series.
1 - (1/1!)/(1*4+1) + (1/2!)/(2*4+1) - (1/3!)/(3*4+1) ...
From the rules of summing a monotone decreasing alternating series {a(n)}, the partial sum up to a(N) is guaranteed to be within |a(N)| of the limit.
Basically, you want (1/N!)/(N*4+1) < 0.001
N=5 should do. It might need more depending on what the 4th correct digit is (if it is 0 or 9 then you might need N=6)
If you're smart, you can get 4 places using only up to N=4.
Doing a second order shanks transform on the first 5 terms gives:
1
4/5, 97/115
77/90, 1217/1440, 630733/746560 = 0.84485
493/585, 26687/31590
67243/79560
