Physics Van 3-site Navigational Menu

Physics Van Navigational Menu

Q & A: mathematical rubbish

Learn more physics!

Most recent answer: 06/16/2014
1 + 2 + 3 + 4 + ⋯=-1/12 This does not seem to make sense. Is this rubbish or correct? I was not convinced by the proof given by the following youtube video. ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 problem/inconsistency? 1) in the video 1-1+1-1+1-1+... = does not give 1/2 but it just fluctuates between 0 and 1 assuming you keep adding +1 and -1 forever in sequence. Why did he conclude that it(=the average value?) is 1/2? What is the average got to do with the actual summation value? problem/inconsistency? 2) in the video the video involves the step adding two identical? infinite series by shifting one and adding them together. but if you shift one, then, it seems no longer the same series or the same operation as adding the two series without shifting. is this not the case? So what gives? 1 + 2 + 3 + 4 + ⋯=-1/12 -> is just baloney? or correct? if correct how so? And more importantly, in what real physical phenomena does this kind of calculation appear (if at all)? (as the physicists in the videos keep implying such phenomena? (calculations?) in physics here and there.) I guess in real physical system there could be two large sets of values (but not sure if they can be infinite sets in physical phenomena) can slide or shift each other but I cannot point to a specific example. Could you provide such examples if possible? As I believe (it seems to me), nature(=physical phenomena, structures?) creates/includes everything(?) including mathematics/physical laws but not the other way around. (it seems mathematics is created by human through observing patterns in nature.) Therefore, I would like to know the answer/explanation from a physics view point but not so much from a mathematics view point. I have seen this appearing in several occasions like: Lawrence M. Krauss || A Universe from Nothing @around 1:14:30 etc. etc.
- Anonymous

You're right that it's rubbish, sophisticated rubbish. I'll answer from a math point of view, because that seems more appropriate for a math problem.

1. As you say the series 1-1+1-1....simply does not converge to any value. There are ways of regularizing such sums that can lead to conversion, but that's a different process, not simple summation. Here, for example, one could multiply the Nth term by exp(-aN), sum, then let the parameter a→0. That sort of recovers the initial sequence, and gives 1/2 for an answer. It's bogus to say the answer is 1/2, however, without including some account of the procedure used to get that.

2. Again, the new series is one that does not converge. If you try to make it converge by using some procedure like the one above, you run into problems because the shift means you're multiplying each part by a different exp(-aN). It's a reminder that you have to define what you're doing, otherwise you can just make up any answer.

3. I believe this is all based on some calculations in which, for some parameter values, the desired function (Riemann zeta) looks just like an infinite sum. If you look at the function for other parameters, you get other values (e.g. -1/12) that are correct for the initial physics problem. Then if you pretend that the function is still an infinite sum for those parameters and write down what the infinite sum would be, as if you could still use the same recipe for translation, you get the nonsense sums of the video.

4. I don't know why Larry Krauss decided to ham it up by peddling the bogus claim. Usually he makes a lot of sense.

Mike W.

(published on 06/16/2014)

Follow-up on this answer.