I invoiced my client for $500 per week, but services were often much more. Now that the project is over, can I invoice him for the unpaid portion?

If there was a letter of engagement or employment, a published rate schedule, weekly time sheets showing the hours worked, and/or a notice on the invoice that your rate or the amount owed was in fact $X, then yes. In that case, the client had notice of your charges and the fact that you did not invoice all of it at once does not affect the client's obligation to pay the full amount.

However, if there was no pre-existing documentation of how much the client should have paid--if you simply submitted invoices for $500 per week--then very possibly not. If the nature of the invoices would lead the client to believe that $500 per week was the full charge, the client is entitled to rely on that. You may not after the fact increase what you are charging them.

So it turns on whether the client had reason to think the $500/week was the full charge, or whether they had notice that the $500 was *not* all they would owe per week.