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I think, rearranging (11+11+11...) as (1+1+1+1...)  (1+1+1+1...) is not valid because of the "shifting" of values, and decomposing of one infinity into two. If you insist on "proving" equality to 0  there is an easier way: just use parenthesis like this: (11)+(11)+(11)... = 0 + 0 + 0 ... which is "clearly" zero. Right? Not really.
This has been bugging me all day, and the best "intuitive" explanation may be based in physics: If you draw (11+11+11...) on a graph assuming it's some physical value over time  it's easy to see that 1/2 is the center of oscillation. So, even though the graph never converges to 1/2  in the infinity it may as well converge. Given the example with a light bulb.. which is ON or OFF, in the infinity, the bulb would be neither ON or OFF  it would be halfbright. If you start sequence with 1, you get an oscillating line around 1/2. So, that makes sense too. If you start doing tricks like (11)+(11)+(11)... = 0 + 0 + 0 ...  it's easy to see that the trick here is selectively collapsing time intervals to 0, which doesn't make sense in the physical sense.
I started today thinking that (11+11+11...) = 1/2 was a fallacy, but now I think it's actually true, and it starts to make sense. However, I'm still to make the leap to understanding how (1+2+3+4+5+...) = 1/12 can be a useful fact, even if it's mathematically correct.