Change user password on the front end
Currently the users/saveUser (and the deprecated users/setPassword) controller action doesn’t support Ajax requests, so there won’t really be an easy way to tell if it worked or not.

If you really want to update user passwords over Ajax, your best bet will be to write a plugin with its own custom controller action, which mimics the password-related stuff in UsersController::actionSaveUser(), but probably calls UsersService::changePassword() rather than saveUser() since you only care about updating the password. And have that controller respond with JSON if it’s an Ajax request, rather than assuming it’s not.

Take a look at EntriesController::actionSaveEntry() for an example of a controller action that does support Ajax.