Proof that some functions do not need to be recursive.

no_recursion.js

//Recursive
function A(n){
    if (n <=0){return 0;}
    return 1 - (1 - A( n - 1)) / 2;
}

//Not recursive, using a loop
function B(n){
    if (n<=0){return 0;}
    //TODO insert smart code here
    answer = 1;
    for (i=0; i<n; i++){
      answer = answer/2;
    }
    return 1 - answer;
}

//Optimized, not recursive, no loop
function C(n){
    if (n<=0){return 0;}
    //TODO insert even smarter code here
    return 1 - (1/(Math.pow(2,n)));
}

var resultString = "<table border=1>";
for (i=0; i<=25; i++){
  resultString += 
      "<tr>" +
      "<td>A(" + i + ")=" + A(i) + 
      "<td>B(" + i + ")=" + B(i) +
      "<td>C(" + i + ")=" + C(i) +
      "</tr>\n";
}
resultString += "</table>";
document.getElementById('result').innerHTML = resultString;

All functions give the same results.