Having an issue with my Moto G starting up.

My phone will not turn on fully after shutting down by itself from a low battery. I came home from work, plugged it in to charge it (which apparently hasn't worked) and what happens is it will turn on, show the start up screen with the Motorola logo on it but it won't even make it to the home screen before it goes black and restarts the process all over again. It doesn't respond to being plugged into a charger and I've tried a normal power up through Fastboot. I'm unsure what to do at this point, wondering if this keeps happening if it is going to cause any permanent damage to the phone or if I should wait until the battery dies completely. Any help would be appreciated. I'm hoping I don't have to replace the phone considering I've had it less than a month.