# Convergents of e

Author: Andrei Osipov

https://projecteuler.net/problem=65

The square root of 2 can be written as an infinite continued fraction.

```√2 = 1 +  1
______
2 +  1
______
2 +  1
______
2 +  1
______
2 + ...```

The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √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 √2.

```1 + 1
___              = 3/2
2

1 + 1
_________       = 7/5
2 + 1 / 2```
`....`

Hence the sequence of the first ten convergents for √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.

Source code: prob065-andreoss.pl

```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)».&take for 2,4 ... * };

say [+] continued-fraction(@e, depth => 100).numerator.comb;

```