Sending a testmail from a server with a configured exim is easy as pie.

From your usershell:

1. tell exim that we want to send a mail to an recipient and tell us about the details of the sending procedure

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   $ exim -v receipient@foo.bar
   

2. Tell exim about the sender, the subject and input some text.

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   From: user@your.domain.example
   Subject: Foobar
   Text Text Text
   
   I like Pie!
   

3. Press Ctrl+d

4. You should see the details about the SMTP connection. You can return to your shell with Ctrl+c