However, at most companies, there's no formal requirement that you do so. Most employment in the United States is called "at will", which means that your current employer can fire you at any time and for any reason (except those that are prohibited under law).
Even if you're not close with your boss, chances are you have a working, first_name_basis relationship. Maintaining that in your official letter is okay. You don't need to keep a Victorian level of formality here__just make sure it's a professional tone.