Problem 81 Hard Difficulty

What is wrong with the following calculation?
$ 0 = 0 + 0 + 0 + \cdot \cdot \cdot $
$ = (1 - 1) + (1 - 1) + (1 - 1) + \cdot \cdot \cdot $
$ = 1 - 1 + 1 - 1 + 1 - 1 + \cdot \cdot \cdot $
$ = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \cdot \cdot \cdot $
$ = 1 + 0 + 0 + 0 + \cdot \cdot \cdot = 1 $
(Guido Ubaldus thought that this proved the existence of God because "something has been created out of nothing.")

Answer

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Video Transcript

Let's determine what's wrong with the following calculation. So here it looks like we have some calculation that's attempting to prove that zero equals one. So this already should raise some red flags. So we're starting off with zero, and then on the right side, we just write. Zero is a bunch of zeros as a song, and then we replace each zero with just one minus one. So there's this one, that one and so on. And then here, from this step to the next steps to the third line there, all we do is removed from disease that's allowed. And then, however, after this step, it looks like we introduced Prentice's again, but we not original Prentice's So I'LL say here introduced Prentice's. So this is a lot because Edition the addition is it's community, I should really say. Here's associative Take a step back here. Edition is associative, So a plus B plus C, we could either at Andy first or if we want we could add B and C first, so here doesn't matter where you put the parentheses, and then after this, it looks like they simplified all of the minus one and plus ones and turn those into zero. However, because we changed the Prentice is we still have this one that pops out, and it looks like the right hand side should be one. So there has to be a mistake here because we know zero cannot equal one. So what's the mistake? Um here. Is that the Siri's over here, for example, you could even write this one right here so we can rewrite. This is one minus one plus one minus one. So it looks like if you look at the theories the way that is, friend, this just says that you're a man is negative one to the end, and then here we should do maybe a plus one. But this thing right here if if this is our end, the limit of a N is not equal to zero. Because if you always equal to negative one or one you'LL never go to zero in the limit. So this will not go to zero. And we can use the divergent test here from your textbook by the test for diversions. This Siri's this one over here. The one that we're working with does not converge. Therefore it makes no sense to say that the Siri's equals one. You cannot equal toe one, if your diversion. So here this is so there's basically two facts here, and they're related. This is the first one, and then the second part is we cannot have diversion Siri's people two one. Therefore, this calculation is bogus because you cannot replace a day but a diversion. Siri's with a number, and that's your final answer.