What is the sum of all positive integers? To clarify: what do you get if you add all the positive full numbers?

1 + 2 + 3 + 4 + 5 + ... + to infinity?

The answer is, of course, in the Internet. What isn't?

Please choose an answer from the list of options. And if interested, look it up on the internet.

Do not post the answer as a comment - until after several days, at least. I'll return to this question after several days and I will respond to some of the comments - if any. I'll also post an answer to the question.

Unlike my previous quizzes in this "Random, silly, fun" category, this question has a definitive answer. (My previous quizzes were "thought experiments" and any answer to them is valid.)

Also based on feedback in my other quizzes, I've added 2 "out" options.

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The answer that I was looking for was indeed "a negative fraction...less than zero but greater than one..." for -1/12. Weird, huh?

This question was based on the Numberphile YouTube video that evidentialist posted below. Numberphile explained how -1/12 can be an answer to the "sum of all positive integers".

Mathologer YouTube channel, some years after, posted a "debunking" video which states that -1/12 is incorrect for pure mathematics. However, Mathologer posted a follow-up video which does confirm that -1/12 can also be a correct answer - in some circumstances.

Both Numberphile's and Mathologer's videos are posted by evidentialist below.

SamKerry
Level 7
Jan 27, 2018

Damn, I voted, thought it was + 1/12, been a while

The answer (which I won't divulge) depends upon whether you are a physicist or not, of course the correct answer can be partially deduced from this comment.

Rufus_Maximus
Level 7
Jan 22, 2018

As a mathematician, the answer to this problem is positive infinity. The sum of positive integers from 1 to n is expressed as the equation n(n+1)/2, which produces {1, 3, 6, 10, 15, ..., +inf}. Taking the limit of n(n+1)/2 as n approaches infinity yields positive infinity.

Now, if the OP asked what is the sum of all integers, then the answer would be zero; 1 + (-1) = 0, or x + (-x) = 0.

Daedalus
Level 4
Jan 22, 2018

>Now, if the OP asked what is the sum of all integers, then the answer would be zero; 1 + (-1) = 0, or x + (-x) = 0.

proofs like that always feel like cheating somehow.

There is no option that supplies the correct answer. The answer is -1/12.

evidentialist
Level 8
Jan 22, 2018

@BeerAndWine -- LOL ... so explain your assertion.

I chose positive infinity as the best of the possible answers. The answer should be a value that approaches positive infinity but never really reaches it. Or so I think.

david7wk
Level 6
Jan 22, 2018

I'll also tag others that replied similarly: @macrobius, @decieven.

I agree that positive infinity is the most logical response. However, "in some circumstances", it actually is -1/12. The video posted by evidentialist, above, presents a good method on how mathematicians came up with -1/12.

The video also mentions that this weird math is used in physics - without giving any actual examples.

I looked on Wikipedia for a use for this. And Wikipedia states "...The regularization of 1 + 2 + 3 + 4 + ? is also involved in computing the Casimir force for a scalar field in one dimension..." My mind went blank. o.O

Without doing any additional digging on the web, it makes sense that it would be positive infinity. It's like the whole concept of different kinds of infinity, depending on how you get there (I don't recall the name off hand). I'm not sure how you could get fractions or negative infinity, though I'd be interested to learn! Some of this deep theoretical math can be intense. A rewarding pursuit though.

Decieven
Level 7
Jan 22, 2018

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