can I find out which generation my iPod classic is by the serial number

For iPods use Identify your iPod model - Apple Support

If you have a later-model iPod Classic you can also find out some detail about it by entering your serial number at Check Your Service and Support Coverage - Apple Support. There seems to be a cut-off for older hardware being included in this support coverage data of about 2010; the iPod Classic range started in 2001, but the last models continued to be available for a few years beyond 2010.

See if this guide helps:

Identify your iPad model - Apple Support

I have an iPod classic, model MC297LL, series 20.5 PC. What generation is this?

[Edited by Moderator]

I believe an iPod Classic running software version 2.0.5 is a 6th generation model.