+1−1+1−1+... Explained in 5 Levels from Algebra to Math Major

Can't believe I'm saying this but, this video gave me nostalgic feeling back when I watch a Numberphile and Mathologer video about this topic

The survey at the beginning was rightfully divided. Resolving the sum is definitely an exercise in "asking a better question" rather than answering the question as stated. Abel and Cesaro are nice generalizations of convergent sums, and adopting either leaves none of the initial ambiguity, but it is a choice and not directly contained in the original question. You also see this with the infamous "-1/12", but usually they're more concerned with blowing your mind than they are with stating the rules of the game clearly. Wallowing in manufactured controversy is more fun than dotting i's and crossing t's.

I am imagining everyone in the comments arguing in a room where no one is listening to each other. fascinating

It's not a convergent series, therefore it may have two answers: 1) It's undefined (because it's not convergent.) 2) It's whatever you want it to be, by making up random definitions, conventions and hand-waving.

In physics class in school i discovered a way to solve it with pascals triangle. We expand the triangle into the negatives -> -1th row is 0+0+0+0+0-1+1-1+1-1+1... Then we take another fact from pascals triangle which is that the n-th row sums up to 2^n. And the negative first row sums up to 1/2

This was a really well put together video and I thought it built up the ideas very well while not providing anything that seemed misleading along the way (by pointing out the inherent weaknesses and problems with various approaches). Cool!

McLovin’ out here dropping knowledge bombs

The best proving: let say that 1<1

The first assumption they made in algebra is that the sum exists

As an engineer, I would ask what the answer is supposed to be used for. Infinite processes don't exist for practical cases; "infinite" is only useful for simplified models. But probability distributions are very useful. If the process is to be sampled or stopped at some random time, the sum would have a 50 % chance of being 0 and a 50 % chance of being 1, with an expectation value of 0.5. Likewise, if you measure voltage in an electrical socket at a random time, in many countries you will measure something between +230 V and -230 V, with an expectation value of 0, i.e. no DC :-)

It’s a non converging sum. You can make it be whatever you want.

In this type of video, I think you should include actual applications of the last two sums, or at least links towards them. Because the Abel sum and the Césaro sum rely on some assumptions and cannot be used everywhere, and the video becomes vacuous without saying what those assumptions are and how these summation techniques satisfying these assumptions and why someone would/can use these techniques.

If you take the series and multiply it by -1, following the same methodology, you see it is equal to -.5 If you do this again, except create a series whose mean is .25, and then subtract this series by the orginal, you get .25 Meaning the rules of Addition and Multiplication seem to be holding. It gets even stranger when you consider this series. +2-1-1.... whose average is 2,1,0 = average of 1 adding this series with the orginal, you get, 3,1,2,0 whose average is 1.5, so even creating wonky random series the methadology holds (as long as you ignore +0s, as you rightfully should) Great video~! The average value of these series is a real mathematical object, so its hard to argue that they aren't 'real' in some way at least. Regardless of calculus saying they dont have a definition.

great video! you mentioned briefly that you used cesaro summation in your research; if you don't mind me asking, where exactly did it come up? what makes it natural to modify the idea behind convergence in this way?

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I would do it like this: You could just add up all the +1 and -1 separately and then calculate the sum, so its ∞ - ∞ which is undefined.

I always think that the 1/2 have a meaning since a lot of the summation methods says it converges to it, if we say that a 1 is representative of a lamp is on (read about Thompson Lamp) and -1 is a lamb is off, and each term can be the condition of the lamb after the nth second. then it makes sense that the time the lamp stayed on is approaching half if multiple timestep occurs. So if we go higher a deminsion and account for time this summation answer will have a meaning on what seems to be a divergent series. Another way to assign a meaning will be using the probabilities, what is the probability that if you randomly observe the lamp that it is on which since the lamp is changing every seconds the probability is 0.5 if we dont know the reference start time

I'd like to see a video that goes deeper into different ways to assign values to divergent series.

I feel like there needs to be another level where techniques related to fourier analysis are used to reduce an infinite series to a succinct description of it.

GREAT VIDEO ❤❤❤❤