This is a case of your iPod Nano simply not holding a charge, which is why it is turning off on you. I understand you have already mentioned that it is not holding a charge either, but that later trait according to Apple is the issue.
Some of the key steps you can initially attempt to resolve this issue, include making sure when you connect your iPod to your PC (or Mac) to make sure the sync cable is securely seated in the iPod and the USB connector is seated in the USB port of your computer.
If you are using Firewire on your iPod, the Firewire port on your PC may not actually be powered, so whenever you connect your iPod, it may be charging the device.
These are some potential troubleshooting steps you can try, however,if you find these few procedures do not work, click the link below to review the remainder of the recommended troubleshooting steps from Apple in addressing charge or power issues with the iPod Nano.
Apple Support Link:
iPod Nano is not chargingLet me know if this helps. If it does not, just let me know and I will be glad to assist further.
Regards,
-Mactechtrainer
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