this infinite sum was featured in a numberphile video a few days ago with a entraining and simple "proof":
http://www.youtube.com/watch?v=w-I6XTVZXww
and a more advanced explanation & proof in the bonus-video: http://www.youtube.com/watch?v=E-d9mgo8FGk
you should definitely subscribe to numberphile (http://www.youtube.com/user/numberphile) and their sister channel computerphile (http://www.youtube.com/user/computerphile) on youtube :)