Ajax assistance please

[Unidentified]

I am trying to update a record in the database with ajax so the page does not reload and it is not posting anything, the php works on its own but not with this ajax :/

    <script type="text/javascript" src="http://code.jquery.com/jquery-latest.js"></script>
<Script>
$(document).ready(function(){
$("#submit").click(function(){
var first_name = $("#first_name").val();
$.ajax({
type: "POST",
dataType: "html",
url: "test.php",
data: "first_name"+first_name,
success:function(data){
 
}
});
})
});
</script>
<form action="" method="post">
<input type="text" class="form-textbox2" id="first_name" size="20"  name="first_name">
<button type="submit" id="submit" name="update"  class="form-submit-button form-submit-button-black_glass">update</button>
</form>

My php which is found in test.php is

<?php
include 'core/init.php';
 
  if( isset($_POST['update']) ) {
 
    try {
 
      $first_name = mysql_real_escape_string($_POST['first_name']);
 
      if( !$first_name ) {
 
        throw new Exception ( '' );
 
      } else {
 
        if( update_first_name($_SESSION['id'], $first_name) ) {
 
          echo 'first name has been updated!';
 
        } else {
 
          throw new Exception ( 'Your first name has been changed successfully .' );
         
 
 
        }
 
      }
 
 
    } catch ( Exception $e ) {
 
      echo $e->getMessage();
 
    }
 
  } else {
 
    echo '';
 
  }
 
 
 
?>