The booklet is the only place where you can find the serial #.
There is a file on your computer, but it is encrypted.
You might be able to get the number from the splash screen (Final Cut Pro > About Final Cut Pro) but I have heard that it is not displayed in full in some releases. It certainly displays in full on the FCP 7 splash screen.
Should none of the above help, follow the instructions here: http://support.apple.com/kb/HT1861