使用Settings API通过AJAX插入选项

Question

I am making a plugin and need to insert a value using the 'add_option()' function - or something that achieves the same result.

This is what I use to call my AJAX:

// localize script
wp_localize_script('button-js', 'the_ajax_script', array('ajaxurl' => admin_url('admin-ajax.php')));

And what launches the PHP function:

add_action('wp_ajax_process_form', array($this, 'process_ajax'));

I need to run add_option() in the function called after my AJAX runs (my AJAX code below):

function post(num) {

    var data = {
        action : 'process_form',
        form_number : num   
    };

    $.post(the_ajax_script.ajaxurl, data).error(
        function() {
            alert('error');
        }).success(function() {
            console.log(data);
        });

        return false;
}   

And here is the function that is run by AJAX:

public function process_ajax() {
    $num = $_POST['form_number'];

    if(isset($num)) {
        add_option('form-num', $num); 
    }

    die();

}

My AJAX code runs successfully, and the correct values are logged to the console - however upon using

<?php echo get_option('form_num'); ?>

The value is blank.