There's an easy way to test the battery.
With the laptop turned off, take the battery out, connect the laptop to the mains adapter/charger, and see if it boots.
If it does then it is likely the battery is dead. Replace with an official ACER approved battery, do not use any cheap batteries.
If it does not boot, then the problem is with the system and not the battery.
Running Vista OS on Acer desktop Model AM5640-B5406A. Shut-down system before bed. Woke up and it was still shutting down, with the small white circle still spinning. I unplugged the computer. After five minutes turned computer back on and nothing happens. It is completely dead. Monitor still works. I do not know what to do. A friend suggested it could be the battery. If I spend $50 and that does not work, I'll feel I have wasted too much money. Should I take it to Office Max? They offer "Free PC Checkup". The store clerk told me they look at it and tell you what is wrong at no charge. Then I can decide if I want it fixed or consider purchasing a new one. Should I trust them? Is there any type of tests I can do at home?