Summary
Sending an email when the invoice is paid is not automatic, but it can be done manually. You can bulk send all invoices created for a given day.
Key Points
Go to Sales > Invoices
Filter the Status as Paid
Select the Invoices
Click the Send to Customer button
From the Dashboard, click Sales
Click Invoices
Change both Date filters to today
Under Status click Paid
Click Filter
Click the check box to the left of the column header Status
Click Send to Customer
To send Invoices out on a weekly basis, change the Date filter to match the weekly criteria in Step 3 and proceed.