*****************Update 12/5/2014*******************************
The battery was not the issue. There was a fundamental issue with the internal surface hardware. It was replaced promptly by Microsoft Surface Support.
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Just a small video about the bugs I am having with the new Microsoft Surface Pro 3. The only thing I did not film was how the Pen sometimes worked and kept on working after I put the tablet to sleep via the button on top, woke it using the top button on the pen, then unlocked the tablet.

That work around seems to always get the pen to work again.

Follow the thread for any fixes on the issue: http://answers.microsoft.com/en-us/surface/forum/surfpro3-surfhardware/recreatable-bug-with-the-surface-pro-3-pen/748aeaf1-d8c7-4071-a52b-0ed3f1e808ff

Please take the opportunity to connect and share this video with your friends and family if you find it useful.