No, the invoices are not automatically sent to the client. This is because the client can view his/her invoices in his / her own profile.
It is possible though to send your client an e-mail with his / her invoice.
Go to the financial app, and search for the relevant client and invoice. After you have opened the invoice you see top right 'options'.
Here you can select 'E-mail'. When you click on it, the invoice is sent by e-mail to the email address of the client listed in the system.