Sum of all positive integers

Originally posted by FabianFnas
In the original Youtube clip they said that 1-1+1-1+... can be either plus one or zero. So 'we set it to be the average = 1/2'. This is BS if anyone ask me. One cannot just reason what the sum is. And this result he uses in the proof of sum of positive integers. Very sloppy, indeed it is.

The sloppiness is all in the explanation. The result of 1/2 is a useful one and not complete BS. But it isn't the true sum as that is defined in every day terms. It is something else.
But I should note here that for any infinite series, even one that converges, one cannot accurately say the sum of the series is a given figure, instead, one says the limit of the series is a given figure.

Originally posted by FabianFnas
When it is about a finite numbers of integers, then the sum is also a integer. So says common sense.
When it is about an in-finite number of integers, then the common sense is all wrong.

Originally posted by twhitehead
Correct.

Okay, then I got it so far. It's the difference of a finite number of terms and an infinite one. You cannot rely the common sense to the sum of infinite number of terms in a series, right. Whatever contra-intuitive result you get it can very well be true anyway.
Let's hold on to this new thinking of mine.

Say you have another sum of an infinite series of terms. Namely 0.33333... .
How many decimals is this? Infinitely many. So is this 1/3 or not. Intuitive it is 1/3 but this new thinking of mine, I'm not sure of this anymore.

s = 0.3 + 0.03 + 0.003 + 0.0003 + ... = sum of 3*10^(⁻i) where i goes from 1 to inf.
s tends to 1/3 when i tends to inf, right. And that's what we have been taught in school. Its intuitive.

But when we do a proof very much as they did in the video - then I'm not sure anymore. IT can be whatever. It can be positive, negative, integer, fraction, non-rational... complex...? The save and sound world of intuitivism has been broken. I don't know anything anymore. This is like an existential crisis.

So what is 0.333... really if you treat it as a truly infinite sum of decimals, rather than an finite series tending to 1/3 ?

Originally posted by FabianFnas
Okay, then I got it so far. It's the difference of a finite number of terms and an infinite one. You cannot rely the common sense to the sum of infinite number of terms in a series, right. Whatever contra-intuitive result you get it can very well be true anyway.

No, what I am saying is that there is no such thing as a sum of an infinite series using the standard definition of 'sum'.
The solution mathematicians use is to either:
1. Use the limit, which IS well defined for series that converge.
2. Use the Riemann zeta function for another set of series that follow a special pattern.
https://en.wikipedia.org/wiki/Riemann_zeta_function

Say you have another sum of an infinite series of terms. Namely 0.33333... .
How many decimals is this? Infinitely many. So is this 1/3 or not.
It is not. It converges to 1/3.
In general, one should avoid ever trying to sum infinities or apply any mathematical operation to them. Instead, always use limits as they can be defined in finite terms.
Playing with infinities will leave you in Zeno territory, and there lies madness:
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

But when we do a proof very much as they did in the video..
The video 'proof' was sloppy at best and outright wrong at worst.

Originally posted by twhitehead
No, what I am saying is that there is no such thing as a sum of an infinite series using the standard definition of 'sum'.
The solution mathematicians use is to either:
1. Use the limit, which IS well defined for series that converge.
2. Use the Riemann zeta function for another set of series that follow a special pattern.
https://en.wikipedia.org/wik ...[text shortened]... they did in the video..
The video 'proof' was sloppy at best and outright wrong at worst.[/b]

Whenever a series converges, then we have no problem.
It is when it diverges where whatever can happen.

I'm not any wiser than before, I'm afraid.
I think I'll stick with the good ol' algebra books from my school days.

But I will have a good time using the -1/2 thingy as a party trick. 😉

Originally posted by FabianFnas
Whenever a series converges, then we have no problem.

Well it seems more intuitive, yes. But it is still wrong to claim you have an actual sum of an infinite series.

It is when it diverges where whatever can happen.
No, 'whatever' does not happen. Only if you get sloppy.
Something similar happens if you play around with dividing by zero.

I think I'll stick with the good ol' algebra books from my school days.
Just stay away from infinity. There be dragons.

sorry, wrong thread.

Originally posted by twhitehead
Just stay away from infinity. There be dragons.

Yes. The rules change. Don't cross the beams!