I'm sorry for posting this late, I just wrote this up as an answer for this (Why is this $0 = 1$ proof wrong?) question before it was marked duplicate. Maybe it has some tiny new bits in it
One possible answer: The proof is wrong because infinite summation is not associative (as is for example shown by this :D )
It is however interesting because it shows that when dealing with infinite series, one has to be careful what one is actually talking about and therefore introduce some notions that might seem technical at a first glance. For example there is usually made a distinction between the sequence of partial sums and it limes.
If the latter one exists (i.e. if the series converge), one sometimes identifies it with it's series, but this identification can lead to problems as in the argument you show: In the first and second line, you are dealing with convergent series, so you can make the identification of limes (left hand side) and sequence of partial sums (right hand side). In the third line, the only way to make sense of the right hand side is a sequence of partial sums, the left hand side is a number however, so we cannot write down equality. The last lines have again a convergent sum on the right, but the sequence of partial sums is different from the one in line two!
I hope this was not too technical, feel free to ask for explanations!
By the way, it is very interesting, that in other contexts, precisely the above argument can be used to show interesting (true) things, for example that you cannot unknot a knot by adding another one to it (check http://en.wikipedia.org/wiki/Eilenberg%E2%80%93Mazur_swindle if you are interested)
Also the proofs of the "sum of natural numbers equals minus one over 12" proofs you mention are probably wrong (Is it the youtube videos?) and i would say not accepted fact (and i really mean the proofs, the statement can still be made sense of in a different way) but still a good way to generate interest for these questions i guess :)