Did you leave the laptop with a dead battery for 3 weeks? Then the battery is probably damaged.
Lithium-ion batteries can catch fire or explode if they are overcharged, or if the are discharged too much and you attempt to recharge them. Consequently, manufacturers build in safeguards which "kill" the battery if either of these conditions happen. If you ran the laptop until the battery died (drained it to 0%), and then left it sitting for 3 weeks, the slow self-discharge rate of the battery probably dropped it below this lower threshold and the battery killed itself to prevent you from causing any physical damage.
In the future, never discharge a battery to 0% (except for calibration). If you do drain it to 0%, recharge it immediately. Set your laptop's backlight to different levels when on battery vs when on AC power. That way you can immediately tell if the AC adapter has stopped working for some reason and you're draining the battery. That'll prevent accidentally draining the battery to 0%.
And I will amend one thing Laptop_Nerd said:
- If the laptop will not boot up with just the AC adapter connected that means the AC adapter is faulty.
It means the AC adapter or the laptop's power port is faulty. If another AC adapter doesn't work as well, your laptop's power port is probably broken, and you'll need to take it to a laptop repair shop.