I have tried the entire day, and can't seem to be able to find out why it is not working. I think that the problem is my skills with JavaScript language. I might be missing some concept on parameter passing. Can you please show me where is the error?
Here is my implementation, based on another implementation I wrote a year ago in C#, which can be found here.
So here is the code (you can try direct on Chromes console, just copy/paste and it will "work"):
function merge(A, p, q, r){
    var n1 = q - p + 1;
    var n2 = r - q;
    var i = 0;
    var j = 0;
    var L = [];
    while (i < n1){
        L.push(A[p + i++]);
    }
    var R = [];
    while(j < n2){
        R.push(A[q + j++ + 1]);
    }
    L.push(Number.MAX_VALUE);
    R.push(Number.MAX_VALUE);
    i = 0; 
    j = 0; 
    var k = p;
    while(k <= r){
        if(L[i] <= R[i]){
            A[k] = L[i];
            i = i + 1;
        }else{
            A[k] = R[j];
            j = j + 1;
        }
        k = k + 1;
    }
}
function mergeSort(A, p, r){
    console.log(A);
    if(p < r){
        var q = Math.floor((p + r) / 2);
        mergeSort(A, p, q); 
        mergeSort(A, q + 1, r);
        merge(A, p, q, r);
    }
}
function testMergeSort(array){
    var p = 0;
    var r = array.length - 1;
    console.log("BEFORE: " + array);
    mergeSort(array, p, r);
    console.log("AFTER: " + array);
    console.log("---------------------------------");
}
testMergeSort([5, 2, 4, 7, 1, 3, 2, 6]);