Will not come on when power source is connected. also, will not come on when power source is not connected

You will need to reset it.
Go to Settings, General, scroll down to Reset.
Ensure you are connected to a power source, select Reset and follow the instructions.
When prompted input your iCloud password and email address.
Restore your data from iCloud or iTunes, whichever you used for your backups.

• Connect your tablet to a power source in the wall by using the wall charger and cable.
• Wait for 10 minutes to make sure that the tablet has gained enough power to start.
• Press and hold the volume down and the power buttons simultaneously for about least 10 to 15 seconds.
• Try to disconnect the battery . This is a common issue with tablets and it can mostly be solved by just disconnecting the battery for a 5 minutes. Or the battery has gone bad and needs to be replaced.

Try turning on while connected to power supply, sometimes after not being used for a while batteries die,they appear to be recharged but are not. If it turns on while on power source you will need a new battery.

Connect to wall outlet Uses the USB power adapter and USB cable included with the device. If it is a third-party accessory, check to see if it is Apple Certified . Then follow these steps:

1. Connect the iPhone, iPad or iPod touch to the Lightning to USB connector cable or the 30-pin to USB connector cable .
2. Connect it to one of three power sources:

Wall outlet

Connect the charging cable to a USB AC adapter and connect the USB AC adapter to the wall outlet.

Computer

Connect the charging cable to a USB 2.0 or 3.0 port on a computer that is not in sleep mode. Do not use the USB ports on the keyboard.

Power supply
Connect the cable to a powered USB hub, docking station, or other Apple-certified accessory.


When the device is charging, you will see a lightning bolt next to the battery icon in the status bar or a large battery icon on the lock screen.

Update you iOS software it's now available! Make sure you have 50% or above on your battery or connected to a power source!

You probably have a loose connection at the power source on the tablet or your power supply is damaged. Have it serviced. If it is a cheap tablet try saving up also for a better tablet like a IPAD. IT IS VERY COMMON TO BREAK THE POWER CONNECTIONS ON TABLETS AND LAPTOPS. GOOD LUCK.

I had a similar problem with my Asus transformer TF101b tablet. First I charged the tablet for three hours. Then while it was still connected to the power source I held down the power button for about 30 seconds. Once I let up on the power button it took about five seconds and the tablet powered on. It took about two minutes for it to boot up and finally get to the home screen.

Sounds like it does not have enough battery power to stay on. If you have not tried to charge it, you should try that first. If you have, it sounds like it is not getting charged or not holding a charge. Could be bad AC power source, bad charger, bad charger connection or bad battery. Take steps to confirm that each of these are good. If all of these are good, sometimes the tablet can become so discharged that it will not allow itself to charge. When they are in that state, sometimes you can trick them into charging again by repeatedly disconnecting and reconnecting the charger as follows: With charger plugged in to AC power source, connect the charger to the tablet. Pay attention to the screen or LED or whatever indicator your tablet has to let you know it is charging. You may see it momentarily indicate charging, or maybe not at all. Then unplug the charger from the tablet then immediately plug it back in and pay attention for the charge indication again. You may have to do this a bunch of times (like 20) until you see the indication that it is charging. When you do get the indication, stop unplugging and re-plugging and let the tablet charge for 24 hours (to a full charge). This trick doesn't always work. Sometimes the battery or charging circuit or an internal connection is bad and no amount of unplugging/re-plugging will fix a hardware problem.

Hi, keep it plugged in to a power source to make sure that it is charged.

Then hold the power button for a few seconds until you see something on the screen.

Last resort would be to take it to the store you got it from to ask them to assist you, and if there is a problem, they should swop it out for you.

Update your device wirelessly
The easiest way to update your device is wirelessly, also called "over the air." Here's how:

• Plug in your device to a power source.
• Tap Settings > General > Software Update.
• Tap Download and Install to download the update. Updates might download automatically while your device is connected to Wi-Fi and a power source.
• Tap Install when the download completes, or tap Later to install the update later.

Update your device using iTunes
If you can't update wirelessly, or if you want to update with iTunes, follow these steps:

• Install the latest version of iTunes on your computer.
• Plug in your device to your computer.
• In iTunes, select your device.
• In the Summary pane, click Check for Update.

Source: http://support.apple.com/kb/ht4623

Hope that helps!