You can't update a psp without a working battery. It'll state that the battery is low and you need to charge it.
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It's only possible to turn on your PSP with an adapter for the PSP or a battery for the PSP.
You can run a PSP without it having a battery, it's not very convenient though. It will only operate while it has the charger in.
Yes ! If you have a PSP 1000 (fat one) like me If you don't have a battery just plug in the charger and it should work.
definitely no the best u can downgrade without a Pandora battery is 3.03.... what you can do is use the Pandora battery to downgrade and the results will be great I've done it on my psp and when i downgraded my psp i got to download as many games that can fit into my psp(full games of course) so downgrade your psp using a Pandora battery:)
You can't, even with the battery.