(I'm verging into stuff I only half-remember here, full disclosure) 1+1/2+1/4+1/8+... is one of the easiest series to just up and say has a solution, that it's 2 Because you're only adding, not subtracting, so you can move around all those numbers at will
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This is called "absolute convergence" If the numbers change sign, you have "conditional convergence" Paradoxically, for people used to doing things a finite number of times, it suddenly matters how the numbers are arranged
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1-2+3-4, if I stop at 4, is always going to give the same result (-2) even if it's -2+3+1-4 or -4-2+1+3 (commutative property of addition) This doesn't work with infinite sums (which doing things an infinite number of times is a filthy lie)
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The classic example is the alternating harmonic series, 1-1/2+1/3-1/4+1/5..., which converges on the number ln(2) If you rearrange the numbers differently, so it's positive+negative+negative and not positive+negative+positive, 1-1/2-1/4+1/3-1/6-1/8+1/5-1/10-1/12... it's ln(2)/2
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Without writing out the proof, you see what kind of dirty trick I'm pulling here, right? For each positive fraction with an odd denominator, I'm "borrowing" an "extra" negative fraction with an even denominator from "the future"
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(Dug up a more in-depth discussion here http://larryriddle.agnesscott.org/series/rearrang.pdf …) If this were a finite set of numbers, then I would eventually run out of "future numbers" to borrow from, and this trick to make the sum constantly be lower wouldn't work, and the sums would come out the same
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But it's an *infinite* series, so I *never* run out And so the same set of numbers rearranged converges to 1/2 the original sum This is why doing things an infinite number of times is, again, a filthy lie
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(Riemann proved you can rearrange this series to converge on ANY sum, at all, or to not converge and instead spit out ∞ or -∞ Which is one of those things mathematicians do that really pisses people off)
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Replying to @arthur_affect
Now I'm shuddering at the thought of them discovering complex numbers. Hell, I've told some people about how you *can* actually take the square root of negative numbers, who just staunchly refuse and go "No, that's nonsense". What more these fanatics of high school math?
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Replying to @GTweetwood @arthur_affect
(Of course, complex numbers *are* nonsense. It's just useful nonsense. As a TA, I usually described it to students as "Tunneling through a wall of nonsense to arrive at a useful result on the other side".
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All numbers are nonsense Have you ever, like, seen a number
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