### The exact moment when it stops being a "sum".

Listen carefully in the explanation video. At one point, the man suggests the series s1 = 1 - 1 + 1 - 1 ... should be taken to be 1/2. The reasoning he gives for this is that "if we stop at an odd point the sum equals 1, if we stop at an even point then the sum equals 0". Both of those are true on their own, but hold on. If we "stop" at a point to observe a value, then that value must just based on that one part of the total sum. We can derive a value by averaging two repeating parts of our sum, but this means that the value is based on the parts, not the sum itself. the fact that the series 1 - 1 + 1 - 1 ... is cyclic (repeats the same pattern of numbers) allowed the presenter to derive a fact about all the parts at once, but this fact was not the sum of the series. The rest of their calculations are all derived from this one, and thus are all based upon facts about the parts that make up the sum, rather than what it equals, or "sums". This doesn't mean that the calculations are useless, though. These numbers represent facts about the sums that could prove useful.