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As stated by Louis and others in their comments, what you need to do is buy a regional ticket that takes you to Cottbus.
You can indeed buy that at TXL airport. There is a DB ticket machines available in a place called Tegel Boulevard according to the TXL website. Unfortunately the linked PDF map doesn't show where that is, but I am sure the information desk will know.
I did a quick search on bahn.de for Flughafen Tevel (Airport), Berlin to Cottbus on 2015-10-30 around 6pm. It says:
It lists the price as 14.30 EUR for a single trip inside the VBB Tarif including the bus and the Regional Express train.