Convergents of e
AUTHOR
Andrei Osipov
https://projecteuler.net/problem=65
The square root of 2 can be written as an infinite continued fraction.
โ2 = 1 + 1
______
2 + 1
______
2 + 1
______
2 + 1
______
2 + ...
The infinite continued fraction can be written, โ2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, โ23 = [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 โ2.
1 + 1
___ = 3/2
2
1 + 1
_________ = 7/5
2 + 1 / 2
....
Hence the sequence of the first ten convergents for โ2 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.
use v6;
sub continued-fraction(@sequence, :$depth) {
my $x = @sequence.shift;
return 1 if $depth == 1;
$x + 1.FatRat /
continued-fraction :depth($depth - 1), @sequence
}
my @e = lazy gather { take 2; (1, $_, 1)ยป.&take for 2,4 ... * };
say [+] continued-fraction(@e, depth => 100).numerator.comb;
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