[exim] exim on raspbian for mail on raspi (via isp -> gmx)

hi everbody,

this might be an easy one for you guys, but please bear with me, since
i'm a little stuck here and everything i found on the net during the
last four hours was a dead end (for me). hopefully you can help.
i want to send mails from my raspberry pi 2 through an email address at
gmx.de (provider). but when i send a test

     echo "Test" | sendmail -v mail@???

all i get is (after successful authentification)

   SMTP>> EHLO raspi
   SMTP<< 250-gmx.com Hello raspi [87.179.83.95]
          250-SIZE 69920427
          250 AUTH LOGIN PLAIN
   SMTP>> AUTH PLAIN ********************************
   SMTP<< 235 Authentication succeeded
   SMTP>> MAIL FROM:<user@raspi> SIZE=1317 AUTH=user@raspi
   SMTP<< 501 Syntax error in parameters or arguments
   SMTP>> QUIT
LOG: MAIN
   ** mail@??? R=smarthost T=remote_smtp_smarthost: SMTP error from 
remote mail server after MAIL FROM:<user@raspi> SIZE=1317 
AUTH=user@raspi: host mail.gmx.net [212.227.17.168]: 501 Syntax error in 
parameters or arguments

i configured /etc/exim4/email-addresses (added /user: //mail@???/),
/etc/exim4/update-exim4.conf.conf (set
/dc_eximconfig_configtype='smarthost'/ and
/dc_smarthost='mail.gmx.net'/) and /etc/exim4/passwd.client (added
/mail.gmx.net:USER:PASS/) accordingly.
i realise that gmx only seem to like fqdns (right?), but it also seems
that it worked without fqdn for other people. also, if the fqdn is
obligatory, is only the syntax important and i can take anything? or do
i need an actual public address/ip? and where do i edit it (i already
tried /etc/hosts and /etc/hostname - but maybe wrongly?)?

thank you very much in advance. if you need more information, i'll be
happy to provide it.

also, i am really, really new to this and hope you can consider this in
your reply. ;)

thanks again and cheers
ken