This has been asked many times in the past and does happen automatically in the background, under specific circumstances:
So it will not be immediate but if an event invite/reminder goes out the new people will get it, also if you use the Newsletter function that will refresh the events and add the new members. If a person asks you can also tell them to select the Globe Icon in the top of the Calendar Window to see all events and not filter by Invited Events, this doesn’t need to be a permanent thing, they can change it back later say after the Newsletter goes out and then they will be invited to all the future events. That doesn’t allow them to RSVP. If you have recently added a member then going into the next couple of calendar events would add them and then future ones would happen automatically once one of the items above happens.