Curious about this

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KevinTang2005

508 posts

KevinTang2005

#1 h Aug 14, 2016, 6:44 PM • 2 Y

Y by Adventure10, Mango247

What is 1+2+3+... (you go on forever)?

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popcorn1

1098 posts

popcorn1

#2 h Aug 14, 2016, 6:50 PM • 1 Y

Y by Adventure10

Theoretically, it should be $-\frac{1}{12}$ . https://www.youtube.com/watch?v=w-I6XTVZXww

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KevinTang2005

508 posts

KevinTang2005

#3 h Aug 14, 2016, 6:50 PM • 1 Y

Y by Adventure10

it can't be -1/12... you're adding a bajillion positive numbers so you should end up with a positive number

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pi_Plus_45x23

2070 posts

pi_Plus_45x23

#4 h Aug 14, 2016, 6:55 PM • 3 Y

Y by rafayaashary1, Adventure10, Mango247

popcorn1 wrote:

Theoretically, it should be $-\frac{1}{12}$ . https://www.youtube.com/watch?v=w-I6XTVZXww

No, theoretically [what does that even mean], the sum diverges.

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Bryanli

839 posts

Bryanli

#5 h Aug 14, 2016, 6:57 PM • 1 Y

Y by Adventure10

$\infty$ duh

This post has been edited 1 time. Last edited by Bryanli, Aug 14, 2016, 6:58 PM

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KevinTang2005

508 posts

KevinTang2005

#6 h Aug 14, 2016, 6:59 PM • 1 Y

Y by Adventure10

Bryanli wrote:

$\infty$ duh

How do you know?

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WireCat

365 posts

WireCat

#8 h Aug 14, 2016, 7:14 PM • 2 Y

Y by Adventure10, Mango247

pi_Plus_45x23 wrote:

popcorn1 wrote:

Theoretically, it should be $-\frac{1}{12}$ . https://www.youtube.com/watch?v=w-I6XTVZXww

No, theoretically [what does that even mean], the sum diverges.

In this case it diverges to $\infty$ , and yes $\infty$ is not a number, but it's a concept of an arbitrarily large number.

popcorn1 wrote:

Theoretically, it should be $-\frac{1}{12}$ . https://www.youtube.com/watch?v=w-I6XTVZXww

With infinite series/sums, you can't do operations such as shifting it over one, as you can easily get things such as $0=1$ , which is why it's a no-no. Something interesting about the formula for the sum is that it has signed area of $\frac{-1}{12}$ underneath the x axis.

This post has been edited 1 time. Last edited by WireCat, Aug 14, 2016, 7:55 PM

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bobjoe123

306 posts

bobjoe123

#9 h Aug 14, 2016, 7:17 PM • 1 Y

Y by Adventure10

https://plus.maths.org/content/infinity-or-just-112

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pi_Plus_45x23

2070 posts

pi_Plus_45x23

#10 h Aug 14, 2016, 7:18 PM • 2 Y

Y by budu, Adventure10

But 1+2+3+... is not equal to -1/12.

This post has been edited 1 time. Last edited by pi_Plus_45x23, Aug 14, 2016, 7:18 PM
Reason: lol typo

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KevinTang2005

508 posts

KevinTang2005

#11 h Aug 14, 2016, 7:19 PM • 1 Y

Y by Adventure10

I don't know how to do calculus, this is probably too hard for me.

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StarFrost7

896 posts

StarFrost7

#12 h Aug 14, 2016, 7:26 PM • 2 Y

Y by oceanair, Adventure10

Don't try to sum this, because it leads to some *really* weird things:

Suppose $S_1 = 1+2+3+4+...$

and suppose $S_2 = 1+1+1+1...$

Note that $S_1$ is self-similar in terms of $S_1:$

$S_1 = 1+S_2 +S_1.$

$S_2 + 1 = 1+1+1+1... = S_2.$

So $S_1 = S_2 + S_1.$

Subtracting $S_1$ from both sides, $0=1+1+1+1+...$ ???

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skipiano

1860 posts

skipiano

#13 h Aug 14, 2016, 7:52 PM • 2 Y

Y by Adventure10, Mango247

The reason we're allowed to do series is because the numbers get so small it doesn't even matter. However, if we shift something like this, say that it is a finite sequence, then there is something large over to the right side that is cut out.

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reaganchoi

5289 posts

reaganchoi

#14 h Aug 14, 2016, 9:19 PM • 1 Y

Y by Adventure10

StarFrost7 wrote:

Don't try to sum this, because it leads to some *really* weird things:

Suppose $S_1 = 1+2+3+4+...$

and suppose $S_2 = 1+1+1+1...$

Note that $S_1$ is self-similar in terms of $S_1:$

$S_1 = 1+S_2 +S_1.$

$S_2 + 1 = 1+1+1+1... = S_2.$

So $S_1 = S_2 + S_1.$

Subtracting $S_1$ from both sides, $0=1+1+1+1+...$ ???

And then, you subtract $1+1+1+...$ from that, except shifted over 1. Subtracting cancels out all the $1$ s except for the leftmost one. This means 0 = 1. QED.

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skywalker1

1204 posts

skywalker1

#15 h Aug 14, 2016, 9:36 PM • 3 Y

Y by StarFrost7, rafayaashary1, Adventure10

here we go again. Look i know there are videos out therethat tell you it is -1/12, but the only one i would even give a little credit to is numberphile. There has been a lot of hate on numberphile in MSM for their video that this sum is -1/12, but in fact the video was made by physicists working in fields where this fact comes n handy. I believ the person in the video has even released a statement(i dont know where this statement is, just that i have heard rumors of its existence and dont doubt its existence either) that the sum does diverge but the fact it is -1/12 is useful in mathematical physics, and modern theories.
For purely mathematical purposes, this series is divergent. Ypu dont even need calculus or even algebra to see this as long as you have a base unerstanding of the divergent sum. A divergent sum doesnt reach a finite value, rather it goes to infinity or -infinity. A third sum which exhibits oscillatory behavior exists, but i donnt think you can call it divergent, althought its is most definitely not convergent. An example of such a sum is 1-1+1-1+1... Dpending on how many 1s in sequence,the sum is either 1 or 0, and it doesnt reach a fixed value.
Now, you could use analytic continuation to get the fact the for people who dont care about divergence, this sum is 1/2, but in reality, its not.
For the other sum of all natural numbers, each time you add a number you get farther away from 0, and you dont approach a real value(inf is not a number). This diverges. You cant say the sum is -1/12. You can say however, that bby analytic continuation, ramanujam summation, or zeta(-1) that we can assign the value of -1/12 to the sum. This serves no purpose other than to reduce the number of times physicists find an infinite number of things from equations, when only a finite number can exist for outr existence; they assign values to infinite sums that do not converge and leave it at that.
Short answer:the sum diverges, its not -1/12
Long answer: if you want to a string theorist, it still diverges but now you have loop holes and it can be -1/12

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fishy15

2219 posts

fishy15

#19 h Aug 15, 2016, 1:18 AM • 2 Y

Y by Adventure10, Mango247

These questions come up a lot

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KevinTang2005

508 posts

KevinTang2005

#24 h Aug 15, 2016, 1:51 AM • 1 Y

Y by Adventure10

I can't find any thread with this? I've scrolled down really far. Can someone give a link to another thread?

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skywalker1

1204 posts

skywalker1

#27 h Aug 15, 2016, 1:58 AM • 2 Y

Y by rafayaashary1, Adventure10

http://www.artofproblemsolving.com/community/c3
http://artofproblemsolving.com/community/c4h1198777p5882415
http://artofproblemsolving.com/community/c3h1139622
http://artofproblemsolving.com/community/c3h1264748
http://artofproblemsolving.com/community/c3h571733
http://www.artofproblemsolving.com/community/c3t31204f3h1101404_sum_of_all_natural_numbers

Took a while but i found a bunch of similar threads if you really want detailed responses

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daniellionyang

1840 posts

daniellionyang

#30 h Aug 15, 2016, 3:42 AM • 2 Y

Y by Adventure10, Mango247

The sum diverges. However, sometimes, mathematicians/physicists assign it to be $-\frac{1}{12}$ to do calculations. The video posted here: https://www.youtube.com/watch?v=w-I6XTVZXww combined with this one: https://www.youtube.com/watch?annotation_id=annotation_2443680779&feature=iv&src_vid=w-I6XTVZXww&v=PCu_BNNI5x4 will shed light on why physicists assign it as $-\frac{1}{12}$ (think, quantum superposition).

This post has been edited 2 times. Last edited by daniellionyang, Aug 15, 2016, 3:59 AM

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