Just like to throw my hat in and say my F710 is also broken. It started with micro-connection lapses, now some buttons go dead sometimes and others stick on, as well as the pad disconnecting and the "mode" light flashing stupidly at me, and refusing to reconnect.
I'm not even one foot from the reciever (I put it on a USB extension lead after it started playing up), and I've tried new batteries.
It failed after 14 months, which is conveniently 2 months outside of warranty. I know for a fact it sat in the packet for a good while before I used it so it's total lifespan has probably been less than one year.
I paid £20 for a new nano reciever, which didn't solve any problems. Now I have to return the reciever on eBay and shop for a new gamepad. It WON'T be a Logitech one.
I'm going to email Logitech and see what they say, although I don't expect anything other than "new batteries">"use connection utility">"be closer to the dongle">"buy a new USB dongle">"you're out of warranty and SOL".
I will report back.
EDIT: So amazingly enough, Logitech offered to RMA my faulty controller even though I purchaes it almost 14 months ago. I explained that I wasn't happy because I hadn't started using it for a while after I bought it so its lifespan had been less than a year, which may be why they have offered to replace it. Equally it could be that they recognise the flashing mode button as a design fault.
Either way, I emailed them proof of postage and my address, and mailed off the controller today (they even paid the postage!). Very very decent of them, after all the could have told me to get lost.
They're a much more helpful company than others I have tried to get help from *cough*ASUS*cough*, I'll update again when I receive the replacement pad 
