My previous post looked at continued fractions and rational approximations for e and gave a little Python code.  I found out later there’s a more direct way to do this in Python using Sage.

At its simplest, the function continued_fraction takes a real number and returns a truncated continued fraction representation. For example, continued_fraction(e)returns

[2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1]

Optionally, you can also specify the number of bits of precision in the real argument and the number of terms desired.

By calling the convergents method on the return value of continued_fraction(e)you can find a sequence of rational approximations based on the continued fraction. For example,

print continued_fraction(e).convergents()

produces

[2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465,
1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089,
517656/190435, 566827/208524, 1084483/398959,
13580623/4996032, 14665106/5394991, 28245729/10391023].

To get higher precision output, you need higher precision input. For example, you could pass in

RealField(200)(e)

rather than simply e to tell Sage that you’d like to use the 200-bit representation of erather than the default precision.