[Puzzler] 1 - 1 + 1 - 1 + 1 - 1...

[Mike]

That's the whole puzzle.

Imagine the mathematical expression 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1... extended on into an infinity of alternating +1's and -1's. What is the value of that expression?

If it helps you to picture it as (+1) + (-1) + (+1) + (-1) + (+1) + (-1) + (+1) + (-1)... then go ahead. It's equivalent and makes some manipulations of the series easier.

[Eliahad]

Or do I have to show my work?

Actually, I don't know the answer, but I'll play around a bit.

[Zen]

Well...



But I could be wrong...

[Eliahad]

But you could also argue that:



Right? So there's something else that I'm missing.

[Zen]

Well... I look at it this way:

But it does seem like there must be something I'm missing...

[Mike]

[DMDarcs]

I was going to say -1/12, but that's something else I'm thinking of.

[Eliahad]

Mike: There is something about your answer that makes it almost certainly correct, but that doesn't sit well with me.


And that's why I became a musician.

[Zen]

According to my son the math major, both Eli and I are right.



But after talking about it for a while, he recalled that you change it to this:



But he was doing this from memory in the kitchen while preparing dinner for himself and his girlfriend, so...

[Eliahad]

Someone get Flameblade in here.

Oh, right that's probably my job.

[Mike]

[Eliahad]

Flame signal has been sent!

[Mike]

Actually, Zen, if you claim that the answer is 0, then you are maintaining that the infinite series ends on a -1. If you think the answer is 1, then you think the infinite series ends on a +1. But it doesn't end on either of those, because it doesn't end.

And you can't use a traditional limit on it, because it doesn't "approach" any number, it just oscillates between 0 and 1.

On the other hand, for cases like this, there is another way to check limits and that's to look at the average of terms (average of first 2 partial sums, then average of first 3 partial sums, average of first 4 answers, etc). When you look at this, the averages of the partial sums:

1, 1/2, 2/3, 1/2, 3/5, 1/2, 4/7, 1/2, 5/9,...

It still oscillates, but the oscillations are clearly approaching 1/2.

People still disagree on the answer, but I like 1/2, because I can actually reach it with maths.

[Eliahad]

Oh, I'm pretty sure you're right.

Just so we're clear, my series STARTS with 1, and then you subtract an infinite series that equals 0 But again, I'm pretty sure you're right.

[DMDarcs]

The YouTube video I watched suggests that 1/2 is the correct answer, but it's not really certain. Mostly because it's not a real limit, but a psuedolimit. Math is weird.

[Zen]

The key is that IT NEVER ENDS. Neither the + or the -. SUM of 1 to infinity of +1 = Positive infinity. Sum of 1 to infinity of -1 = Negative infinity. Positive infinity + Negative infinity = ?

I'm sure if you do the calculus stuff on it, it will totally come up with something different.

[FlameBlade]

Mike got it. It's 1/2.

Specifics is close to what Mike is claiming.

First, is the sequence diverging? No. This is the important bit. But the sequence does not converge either, but it is bounded.

Yes, it's such a disgusting answer, as it is also arguable that it is either 0 and 1, but yet at the same, it is neither because sum do not end!

More explanation here:. https://en.m.wikipedia.org/wiki/Grandi%27s_series

[Zen]

I still don't understand how positive infinity + negative infinity = 1/2... but I guess that's because infinity sucks.

Leibniz answer was worse! He came up with the same thing, but it was basically a massive kluge.

EDIT: and Daniel Bernoulli somehow got 2/3?

[Mike]

Here's the video I pulled everything here from:



So I get no credit for smarts here except for parroting what a YouTube video told me. But I'm very fond of this problem now in the same way that I love explaining to novices how 0.9999999... = 1.

[Mike]

And thank you for responding to the Bat-Signal, Flame. Out of respect, we're going to make you your own Flame-Signal for any future math emergencies.