If you just got a brand new iDevice or just re-flashed your existing iDevice, Gmail push using the Exchange account will no longer work. It will keep on giving connection errors to the server and asking for your password over and over again.
This is because Google stopped it’s ActiveSync service for free accounts from the end of January 2013. Check out this link for a full description of the same.
So, if you are looking for Gmail push notifications, check out alternatives like IM+ Pro, Mailbox and the likes. In case you are feeling really adventurous, you could even try out the Google GMail app for iOS.