Paid Memberships Pro emails a membership invoice for all recurring or renewing membership subscriptions. This email is not sent after the initial membership checkout because all of the relevant invoice details are included in the Membership Confirmation email. If you’ve customized the Confirmation email and removed invoice details, you may want to send a separate…
If you’d like better information about how system-generated emails are being received, read, clicked or bounced, you can configure Mandrill as your outgoing mail service. Mandrill harnesses the delivery rates of MailChimp and overrides the wp_mail function included with WordPress. This method can be used for sites with and without Paid Memberships Pro.
This is also useful for troubleshooting email delivery issues as explained here.