Ajax for Plugins

I have the code in a file save_author_description.js:

$(document).ready(function() {
        $('#save_edit_author_description').click(function(){
        var description = $('#edit_author_description').val();
        console.log(description);
          	$.ajax({
            type: 'POST',
            url: '/wp-admin/admin-ajax.php',
            data: {
                action: 'update_user_description',
                description: description,
            },
            success: function(data){
                $('#author_description_saved').html('').append(data);
            },
            error: function(){
                alert('error');
            }
        });
    });
});

in a js file. And in functions.php I have :

function add_save_author_description(){
	wp_enqueue_script('save_author_description.js', get_bloginfo('template_directory'). "/scripts/save_author_description.js", array('jquery'));
}
add_action('init','add_save_author_description');

function update_user_description(){
	global $wpdb;
	$description = $_POST['description'];
	if(isset($description)){echo $description;}else{echo 'nothing';}
	die($description);
}
add_action( 'wp_ajax_nopriv_update_user_description', 'update_user_description' );
add_action( 'wp_ajax_update_user_description', 'update_user_description' );

and html of:

<textarea style="display:none;" id="edit_author_description">Tell Us a Little About Yourself</textarea>
	<button style="display:none;" id="save_edit_author_description">Save</button>
	<div id="author_description_saved"></div>

However, when I click on the button with id of ‘save_edit_author_description’ literally nothing happens. Any ideas?