My iPhone 3 will not turn on? - 1

The apple logo will go up and then it starts to power off but with the apple logo still on and then it flashes white and starts again and when its dead but o charge it will try to turn on and then it will connect to iTunes but then disconnect. It has powered on a few times and it works fine. It does have cydia but it is out of date so i don't know how that could be effecting the phone but I've never jail broken a phone before so i don't know. It does have cracks but phones have cracks on them all the time. Can it be fixed? I don't want to fix it myself because i have no experience with phones at all, and where can i get it repaired? I live in Brisbane (Australia) and i can't find a single place that won't charge me a thousand dollars and I don't even know what's wrong with it. Please answer.

Added (1). it just completely turned off now and its not doing anything i think its dead

Take it to apple

Do you still have your service?