I would be concerned too if my iPad was not powering on. I recommend following the steps below to address an issue like this (these steps refer to an iPhone, but the steps also are applicable to the iPad):
Will not turn on, will not turn on unless connected to power, or unexpected power off
- Verify that the Sleep/Wake button functions. If it does not function, inspect it for signs of damage. If the button is damaged or is not functioning when pressed, seek service.
- Check if a Liquid Contact Indicator (LCI) is activated or there are signs of corrosion. Learn about LCIsand corrosion.
- Connect the iPhone to the iPhone's USB power adapter and let it charge for at least ten minutes.
- After at least 15 minutes, if:
- The home screen appears: The iPhone should be working. Update to the latest version of iOS if necessary. Continue charging it until it is completely charged and you see this battery icon in the upper-right corner of the screen . Then unplug the phone from power. If it immediately turns off, seek service.
- The low-battery image appears, even after the phone has charged for at least 20 minutes: See "iPhone displays the low-battery image and is unresponsive" symptom in this article.
- Something other than the Home screen or Low Battery image appears, continue with this article for further troubleshooting steps.
- If the iPhone did not turn on, reset it while connected to the iPhone USB power adapter.
- If the display turns on, go to step 4.
- If the display remains black, go to next step.
- Connect the iPhone to a computer and open iTunes. If iTunes recognizes the iPhone and indicates that it is in recovery mode, attempt to restore the iPhone. If the iPhone doesn't appear in iTunes or if you have difficulties in restoring the iPhone, see this article for further assistance.
- If restoring the iPhone resolved the issue, go to step 4. If restoring the iPhone did not solve the issue, seek service.
You can find the full article here:
You can also check the service options for the iPad here (you can change the country at the top of the page):
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I just want to confirm you are using the original iPad charger and not a charger for an iPod or iPhone (either apple or third party). IPads require 10w of power to charge and standard iPhone/iPod chargers only output 5w. On the question of battery replacement what they do is just swap out the ipad for "new" reconditioned one. I'm not sure of the cost but if your iPad is a few years old perhaps you might want to determine if it would be better value to get a new one.