4kyu_MagnetParticulesInBoxes.js

// 4kyu - Magnet Particules in Boxes

// Professor Chambouliard hast just discovered a new type of magnet material. 
// He put particles of this material in a box made of small boxes arranged in K rows 
// and N columns as a kind of 2D matrix K x N where K and N are postive integers. 
// He thinks that his calculations show that the force exerted by the particle in 
// the small box (k, n) is:

// v(k, n) = \frac{1}{k(n+1)^{2k}}

// The total force exerted by the first row with k = 1 is:

// u(1, N) = \sum_{n=1}^{n=N}v(1, n) = 
// \frac{1}{1.2^2} + \frac{1}{1.3^2}+...+\frac{1}{1.(N+1)^2}

// We can go on with k = 2 and then k = 3 etc ... and consider:

// S(K, N) = \sum_{k=1}^{k=K}u(k, N) = 
// \sum_{k=1}^{k=K}(\sum_{n=1}^{n=N}v(k, n)) \rightarrow (doubles(maxk, maxn))

// #Task: To help Professor Chambouliard can we calculate the function 
// doubles that will take as parameter maxk and maxn such that doubles(maxk, maxn) = 
// S(maxk, maxn)? Experiences seems to show that this could be something around 0.7 
// when maxk and maxn are big enough.

// #Examples:

// double(1, 3)  => 0.4236111111111111
// double(1, 10) => 0.5580321939764581
// double(10, 100) => 0.6832948559787737

function doubles(maxk, maxn) {
    let fn = (a,b) => 1/(k * Math.pow((n + 1), 2*k)), k = 1, s = 0;
    while(k <= maxk) {
      for(var n = 1; n <= maxn; n++) 
        s += fn(k,n);
      k++;
    }
    return s;
}