The invoices you send to your customer will not sync with your personal email account. You'll want to add your email address in the Cc, Bcc fields, or select the Email me a copy option so you'll also receive any invoice you send to them.
Here's how:
Go to the Gear icon and select Account and Settings.
From the Sales menu, go to the Messages section.
You can check the Email me a copy at option and enter your email address in the Copy (Cc) or Blind copy (Bcc) fields.
Click Save and then Done.
To help you manage your customer transactions, you can check out these articles: