JSFiddle - React, Tailwind, and code Playground

HTML

<style>
  body {
    font-family: monospace;
    font-size: 14px;
    margin: 30px;
  }

</style>

<body>
  <script>
    // https://physics.nist.gov/cgi-bin/cuu/Value?alph
    const alpha = 0.0072973525693;

    function log(text) {
      var div = document.createElement("div");
      div.innerHTML = text;
      document.body.appendChild(div);
    }

    // calculate series to n terms. can edit n. converges fast.
    function calc_gamma(alphaval) {
      var gammaval = 1;
      for (var n = 10; n >= 0; n--) {
        gammaval = 1 + (alphaval / (Math.pow(2 * Math.PI, n))) * gammaval;
      }
      return gammaval;
    }

    // iterate i times. can edit i. converges fast.
    function iterate_alpha(ansatz) {
      var gammaval;
      var alphaval = ansatz;

      for (var i = 0; i < 10; i++) {
        gammaval = calc_gamma(alphaval);
        alphaval = Math.pow(gammaval, 2) * ansatz;
      }
      return alphaval;
    }

    // e^(- pi^2 / 2)
    var ansatz = Math.pow(Math.E, -1 * (Math.PI * Math.PI) / 2);
    var alphavalout = iterate_alpha(ansatz)

    log('NIST value');
    log(alpha);

    log('<br>');

    log('Iterated value');
    log(alphavalout);

    log('<br>');

    log('Ratio');
    log(alpha / alphavalout);

  </script>
</body>