1.1 KiB
1.1 KiB
The infinite continued fraction can be written, sqrt(2) = [1;(2)], (2) indicates that 2 repeats ad infinitum.
In a similar way, sqrt(23) = [4;(1,3,1,8)].
It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations.
Let us consider the convergents for sqrt(2).
1 + 1/2 = 3/2
1 + 1/(2+ 1/2) = 7/5
1 + 1/(2+ 1/(2+ 1/2)) = 17/12
1 + 1/(2+ 1/(2+ 1/(2+ 1/2))) = 41/29
Hence the sequence of the first ten convergents for sqrt(2) are:
1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...
What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...]
The first ten terms in the sequence of convergents for e are:
2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...
The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.
Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.