Your serial number is not erased by uninstalling Lightroom. The serial number from the Beta is stored in a file on your computer, and it may be valid for you full version of Lightroom, but it will expire and give you trouble later.
The best thing to do is go to Help > Lightroom Registration. Likely, you will see name and address fields. Ignore this and look at the left hand column:
Licese or Try
Go to License and you will have the boxes to enter your serial number. Simply enter your serial number and click "Finish".