data/problems/65.yml
---
:id: 65
:name: Convergents of e
:url: https://projecteuler.net/problem=65
:content: "The square root of 2 can be written as an infinite continued fraction.\n\n|
√2 = 1 + | \n1\n |\n| | 2 + | \n1\n |\n| | | 2 + | \n1\n |\n|
| | | 2 + | \n1\n |\n| | | | |
2 + ... |\n\nThe 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)].\n\nIt 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.\n\n|
1 + | \n1\n | = 3/2 |\n| | \n2\n | |\n\n| 1 + | \n1\n | = 7/5 |\n|
| 2 + | \n1\n |\n| | | \n2\n | |\n\n| 1 + | \n1\n |
= 17/12 |\n| | 2 + | \n1\n | |\n| | | 2 + | \n1\n |
|\n| | | | \n2\n | |\n\n| 1 + | \n1\n | = 41/29
|\n| | 2 + | \n1\n |\n| | | 2 + | \n1\n | |\n|
| | | 2 + | \n1\n | |\n| | | |
| \n2\n | |\n\nHence the sequence of the first ten convergents for √2 are:\n\n1,
3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...\n\nWhat
is most surprising is that the important mathematical constant, \n_e_ = [2; 1,2,1,
1,4,1, 1,6,1 , ... , 1,2_k_,1, ...].\n\nThe first ten terms in the sequence of convergents
for _e_ are:\n\n2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536,
...\n\nThe sum of digits in the numerator of the 10<sup>th</sup> convergent is 1+4+5+7=17.\n\nFind
the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued
fraction for _e_.\n\n"