Personally, I encountered this in the wild. My brother asked me “do you know what the series 1 + 2 + 3 + 4 + 5 + and so on sums up to?” “Well,” I said, “That sums up to infinity.”
“No, it’s −1/12!”, he exclaims. I exclaimed that this was bullshit—there are only positive numbers in the series, there are only additions in the series, and since adding positive numbers together produces a positive number, the “negative a twelfth” result is just plain wrong.
We had a bit of an argument after that, after which he said “yes, you’re right, but when you sum them up ALL AT ONCE then you get negative a twelfth”. And I accepted that, because, well, summing them up all at once is a different operation than repeated addition, so then you might get some different result. I didn’t study complicated maths, I don’t know what happens when you wrap things with complicated functions, and I’m willing to concede that there exists some fancy way of wrapping repeated addition in a mathematical construct so that you can get a negative result.
So this post comes along, and yes. You take a fancy mathematical function and apply it over the repeated addition, just as expected.
Personally, I encountered this in the wild. My brother asked me “do you know what the series 1 + 2 + 3 + 4 + 5 + and so on sums up to?” “Well,” I said, “That sums up to infinity.”
“No, it’s −1/12!”, he exclaims. I exclaimed that this was bullshit—there are only positive numbers in the series, there are only additions in the series, and since adding positive numbers together produces a positive number, the “negative a twelfth” result is just plain wrong.
We had a bit of an argument after that, after which he said “yes, you’re right, but when you sum them up ALL AT ONCE then you get negative a twelfth”. And I accepted that, because, well, summing them up all at once is a different operation than repeated addition, so then you might get some different result. I didn’t study complicated maths, I don’t know what happens when you wrap things with complicated functions, and I’m willing to concede that there exists some fancy way of wrapping repeated addition in a mathematical construct so that you can get a negative result.
So this post comes along, and yes. You take a fancy mathematical function and apply it over the repeated addition, just as expected.