Q: iPhone 5 stuck in recovery mode and wont restore on iTunes

Here's a link to the Apple support page on error 4005:

iOS: Restore errors 4005, 4013, and 4014 - Apple Support

That error seems to be related to a need to update the version of iTunes on your computer.

I know you say you have the latest version, but if you go into iTunes, what version does it say you have? Also, what computer OS are you using? Is it up to date? If you have all of that up to date, then as the support document says, contact Apple Support.

That is the latest iTunes. I don't doubt you, but many times people say they have the latest but have never checked. Thank you. If your Windows is up to date, there is no other alternative in the support than to contact Apple with that error. Make an appointment at the Genius Bar of the nearest Apple store.