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 + 1/2 = 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 for the first ten convergents of √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 convergents of 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 the digits in the numerator of the nth convergent of the continued fraction for e.
Information at Project Euler 065
Getting Started
Enter a whole number between 2 and 100 (without leading zeros) in the input field and select the Submit Button. You will see the sum of digits in the numerator of the nth convergent of the continued fraction of e, unless you have made an invalid input. Select the Reset Button to clear the information or to start again.
User Stories
As a user, I expect to get an error message, if I do any of:
- Not enter anything in the input field
- Entering text other than a number
- Entering a number less than 2 or greater than 100
- Including leading zeros such as 02
- Entering a number, but it is not an integer
As a user, if I select the Reset Button, I can clear the information or start again.
As a user, I expect the function convergentsOfE(10) to return a number.
As a user, I expect the function convergentsOfE(10) to return 17.
As a user, I expect the function convergentsOfE(30) to return 53.
As a user, I expect the function convergentsOfE(50) to return 91.
As a user, I expect the function convergentsOfE(70) to return 169.
As a user, I expect the function convergentsOfE(100) to return 272.
User Stories on function convergentsOfE(n) taken from FreeCodeCamp - Coding Interview Prep - Project Euler 065
Information Architecture
The function convergentsOfE(n) returns a number, where n is a number between 2 and 100.
Allows the user to enter a number in order to get the sum of digits in the numerator of the nth convergent of the continued fraction for e. Performs checks on valid user input. If the input is invalid, an error message is displayed.
Uses HTML5, CSS3, JavaScript with BigInt, Bootstrap 5.2.3 and Google Fonts.
Ensure all user stories have been met.
Deployed on GitHub Pages at the main branch.
Written by me.