My iPhone won’t turn on…Any tips?

My iPhone keeps displaying the Apple logo when I press the power button, but automatically shuts off after. I got my iPhone 3g a little over a month ago. I fully charged it last night, and was even using it this morning. However, when I just checked on it, I found it turned off. I pressed the power button, and the apple logo showed up. Then it turned off again. My iPhone won’t stay on. I’ve tried resetting it to no avail. I plugged it into my PC, and the computer won’t even recognize it. The iPhone is practically new…I’ve never dropped it, it hasn’t been water damaged and the screen is perfectly intact. I don’t want to take it to the Apple Store because it’s a good distance from where I live…Any suggestions?