/sci/ - Science & Math » Thread #14254294

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Anonymous Tue 22 Feb 00:07:36 2022 No.14254294 View Reply Original Report

Quoted By: >>14254296 >>14254301 >>14254392 >>14254443 >>14254526 >>14254580 >>14255205 >>14255293 >>14255304 >>14255310 >>14256849

well?

Anonymous

Anonymous Tue 22 Feb 2022 00:08:34 No.14254296 Report

Quoted By: >>14254532 >>14254978 >>14255205 >>14256866 >>14258358

>>14254294
Uh huh, that's why you can't manipulate infinite sums like that.

Anonymous

Anonymous Tue 22 Feb 2022 00:11:09 No.14254301 Report

Quoted By:

>>14254294
It's technically a true statement in situations involving analytic continuation, otherwise nah.

Anonymous

Anonymous Tue 22 Feb 2022 00:37:16 No.14254392 Report

Quoted By:

>>14254294
I just had an interesting idea

starting from here
1 + 2 + 4 + 8 + 16 + ... = -1

if you subtract each term on the left side term by term, the right expression term follows the sequence S

- 1 , - 2 , - 3 , - 7 , - 15 , - 31 , - 63 , ... = S

So Sn after n = 1 is always the opposite of yn + 1

Anonymous

Anonymous Tue 22 Feb 2022 00:52:03 No.14254438 Report

Quoted By: >>14256866 >>14258358

you can't do algebra with the limit of sequences like 1 + 2 + 3 + ... + n unless you prove that the limit exists.

Anonymous

Anonymous Tue 22 Feb 2022 00:53:19 No.14254443 Report

Quoted By:

>>14254294
what are partial sums: the post

Anonymous

Anonymous Tue 22 Feb 2022 01:12:54 No.14254497 Report

Quoted By: >>14254708

S = ... + a^-2 + a^-1 + 1 + a + a^2 + ...

aS = ... + a^-1 + 1 + a + a^2 + a^3 + ...
= ... + a^-2 + a^-1 + 1 + a + a^2 + ...

aS = S

S = 0

The sum of all the powers of a number is 0

Anonymous

Anonymous Tue 22 Feb 2022 01:23:21 No.14254526 Report

Quoted By: >>14254532 >>14255205

>>14254294
infinity = 1 + 2 infinity
Can't do that.

Anonymous

Anonymous Tue 22 Feb 2022 01:28:00 No.14254532 Report

Quoted By:

>>14254296
>>14254526
OP here. these are the correct answers.

congrats /sci/, you passed.

Anonymous

Anonymous Tue 22 Feb 2022 01:45:10 No.14254564 Report

Quoted By:

Nibbas don’t know what an indeterminate form is

Anonymous

Anonymous Tue 22 Feb 2022 01:49:18 No.14254580 Report

Quoted By: >>14254592

>>14254294
as a corollary, 1 + 3 + 5 + 7 + ... = -1/12

Anonymous

Anonymous Tue 22 Feb 2022 01:53:38 No.14254592 Report

Quoted By: >>14258567

>>14254580
also
1 + 2 + 3 + ...
=(1 + 2 + 4 + 8 + ...) + 3(1 + 2 + 4 + ...) + 5(1 + 2 + 4 +...) +...
=-1(1 + 3 + 5 + ...)
=-1(2*(1 + 2 + 3 + ...) + (1 + 1 + 1 + ...)
so -1/12 = -1(-1/6 + x) where x = 1 + 1 + ...
-1/12 = 1/6 - x
1 + 1 + 1 + ... = 1/4, QED

Anonymous

Anonymous Tue 22 Feb 2022 01:55:31 No.14254597 Report

Quoted By: >>14254720 >>14256321 >>14256822 >>14258567

x = 1 + 4 + 16 + 64 + 256 + ...
x = 1 +4(1 + 4 + 16 + 64 + ...)
x = 1 + 4x
x = (-1/3)

z = 1 + n(1 + n + n^2 + n^3 + ...)
z = 1 + nz
-(n-1)z = 1
z = -1/(n-1)

If we let n -> infinity then we get
a= 1 + (infinity)(1 + (infinity + (infinity^2) + (infinity^3) + ...)
a = 1 +(infinity)a
a = 0

Gibberish math is fun

Anonymous

Anonymous Tue 22 Feb 2022 02:56:20 No.14254708 Report

Quoted By:

>>14254497
a could be 1 too

Anonymous

Anonymous Tue 22 Feb 2022 03:01:06 No.14254720 Report

Quoted By: >>14256822

>>14254597
>-(n-1)z = 1
where is that negative sign on the left side coming from?

>z = -1/(n-1)
that 1 shouldn't have a negative sign but this seems interesting

Anonymous

Anonymous Tue 22 Feb 2022 03:02:44 No.14254727 Report

Quoted By:

S=1+1+1+...
S=1+S
0S = 1
0 = 1

Freshman BTFO

Anonymous

Anonymous Tue 22 Feb 2022 03:50:23 No.14254822 Report

Quoted By: >>14254836 >>14254850

If we have two series:
a: 1, 2, 3, 4, 5 . . .
b: 2, 4, 6, 8, 10 . . .

Can it be said that while sums of both a and b are infinite, the sum of b is also two times the sum of a?

Anonymous

Anonymous Tue 22 Feb 2022 03:57:23 No.14254836 Report

Quoted By: >>14254867

>>14254822
If we call each partial sum $a_n$ and $b_n$ , then $\lim_{n\to\infty} \frac{a_n}{b_n} = \frac{1}{2}$ . However, $\frac{\lim_{n\to\infty} a_n}{\lim_{n\to\infty}b_n}$ does not exist.

Anonymous

Anonymous Tue 22 Feb 2022 04:05:37 No.14254850 Report

Quoted By: >>14254867

>>14254822
>Can it be said that while sums of both a and b are infinite, the sum of b is also two times the sum of a?
It can be said. But it’s wrong.

Anonymous

Anonymous Tue 22 Feb 2022 04:15:19 No.14254867 Report

Quoted By: >>14254971

>>14254836
So, in layman's terms because I'm not very bright, would it mean that while the sum of b is twice the sum of a in this exact (with the implied element-to-element relation) comparison, the sums of the actual independent infinite series cannot be compared?

>>14254850
Could you please elaborate?
I thought that'd be something with an easily searchable explanation but I found nothing.

Anonymous

Anonymous Tue 22 Feb 2022 04:39:30 No.14254916 Report

Quoted By:

Literally ramanujan math.

Anonymous

Anonymous Tue 22 Feb 2022 05:13:28 No.14254971 Report

Quoted By: >>14254979 >>14255090

>>14254867
Pretty much, comparing them term by term places a restriction on how fast one can grow relative to the other. However, when we think about the limits separately, no such restriction exists.

For sequential limits that actually exist, this is never an issue. That is, if $a_n$ and $b_n$ are two sequences converging to $A$ and $B$ , respectively, then $\frac{\lim_{n\to\infty} a_n}{\lim_{n\to\infty} b_n} = \frac{A}{B}$ . This holds true for any subsequence as well.

Going back to our original example, if we compare a subsequence $a_{2n}$ to $b_n$ , we find that $\lim_{n\to\infty}\frac{a_{2n}}{b_n} = 2.$

Anonymous

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Anonymous Tue 22 Feb 2022 05:19:08 No.14254978 Report

Quoted By: >>14255207

>>14254296

Anonymous

Anonymous Tue 22 Feb 2022 05:19:51 No.14254979 Report

Quoted By: >>14255090

>>14254971
Should not that B can't equal two zero either, but hopefully that was already understood.

Anonymous

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Anonymous Tue 22 Feb 2022 06:08:24 No.14255048 Report

Quoted By:

>Why yes, this is perfectly valid in the field of 2-adics

Anonymous

Anonymous Tue 22 Feb 2022 06:57:17 No.14255090 Report

Quoted By:

>>14254971
>>14254979
That's very cool, anon, thanks for explaining.
Something I'd like to confirm - when you're talking about
>a2n/bn = 2
shouldn't it be 1 instead?
Or am I missing something?

Anonymous

Anonymous Tue 22 Feb 2022 08:58:03 No.14255205 Report

Quoted By: >>14255208

>>14254294
This is correct
>>14254296
Indoctrinated
>>14254526
Infinity isn't a number.

Anonymous

Anonymous Tue 22 Feb 2022 09:00:02 No.14255207 Report

Quoted By:

>>14254978
You did something that doesn't make sense, and got a result that doesn't make sense, just like they warned you. Good job.

Anonymous

Anonymous Tue 22 Feb 2022 09:01:14 No.14255208 Report

Quoted By: >>14255216

>>14255205
>you're le heckin' indoctrinated!!
For knowing that carelessly mucking around with diverging sums leads to contradictions?

Anonymous

Anonymous Tue 22 Feb 2022 09:04:00 No.14255213 Report

Quoted By: >>14255336

1+1+1+...=1+(1+1+1+...)
s=1+s
1=0
I'm so smart.

Anonymous

Anonymous Tue 22 Feb 2022 09:05:38 No.14255216 Report

Quoted By: >>14255241

>>14255208
It doesn't. The theory of divergent series is completely consistent.

Anonymous

Anonymous Tue 22 Feb 2022 09:20:10 No.14255241 Report

Quoted By: >>14255247

>>14255216
>b-b-but m-m-muh theory of divergent series
At that point, you're not even dealing with real sums anymore.

Anonymous

Anonymous Tue 22 Feb 2022 09:24:24 No.14255247 Report

Quoted By: >>14255253

>>14255241
>At that point, you're not even dealing with real sums anymore.
Indoctrinated.

Anonymous

Anonymous Tue 22 Feb 2022 09:28:14 No.14255253 Report

Quoted By: >>14255312

>>14255247
You can use the same mouth noise to reference them, but they don't work like real sums. Case closed.

Anonymous

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Anonymous Tue 22 Feb 2022 10:18:31 No.14255293 Report

Quoted By:

>>14254294
So you have an expression that follows some kind of internal logic, but you can't directly relate it to any normal value, so you find a way to relate it to itself, and discover that it relates to itself the same way -1 relates to itself. Good job, you've found a long-winded way to say "-1". This doesn't actually tell you anything about addition, because there is obviously no relationship between what happens when you try to actually add those numbers together, and the value you can assign to it if you treat it as a special object that can be manipulated using some special rules that reflect its internal logic.

Anonymous

Anonymous Tue 22 Feb 2022 10:34:50 No.14255304 Report

Quoted By: >>14255312

>>14254294
? = 2? + 1
And that is pretty much why ? is not exactly a number.

Anonymous

Anonymous Tue 22 Feb 2022 10:40:25 No.14255310 Report

Quoted By:

>>14254294
>y = infinity
>y = 1 + 2 * infinity
>y = 1 + 2y

Anonymous

Anonymous Tue 22 Feb 2022 10:42:29 No.14255312 Report

Quoted By: >>14255329 >>14255386

>>14255253
The sum of a series is the constant term of the smoothed asymptote. There is nothing different about sums of divergent series.
>>14255304
>Infinity isn't a number
>The sum of the series is infinity
Brain damage

Anonymous

Anonymous Tue 22 Feb 2022 10:55:59 No.14255329 Report

Quoted By:

>>14255312
>The sum of a series is the constant term of the smoothed asymptote
Like I said, you can imbue the same mouth noise with a different, generalized meaning that violates the properties of the normal, but you're simply engaging in ackchually-tier autism.

Anonymous

Anonymous Tue 22 Feb 2022 10:58:08 No.14255336 Report

Quoted By: >>14258409

>>14255213
I don't see the issue there. You determined that no value can be consistently assigned to your sum. This doesn't invalidate OP's sum.

Anonymous

Anonymous Tue 22 Feb 2022 11:29:43 No.14255386 Report

Quoted By: >>14255387 >>14255408 >>14256512

>>14255312
>The sum of the series is infinity
Yep. And it's not exactly a number, because it's literally innumerable, you cannot tell if it's even or odd, for example. It is a concept, a more abstract concept than even numeral.

Anonymous

Anonymous Tue 22 Feb 2022 11:30:45 No.14255387 Report

Quoted By: >>14255408

>>14255386
I have to concur with him that you have brain damage.

Anonymous

Anonymous Tue 22 Feb 2022 11:55:58 No.14255408 Report

Quoted By:

>>14255386
>you cannot tell if it's even or odd, for example.
in that infinite "not a numeral" you actually can, it's always odd. Weird. Okay, but once again..
? = 2? + 1
doesnt' the second half grow twice as fast?
>>14255387
I'm just not a mathematician, I'm sorry for bothering you, numerologists (in a good sense)

Anonymous

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Anonymous Tue 22 Feb 2022 18:20:27 No.14256321 Report

Quoted By:

>>14254597
I'm getting combinatorics flashbacks.

Anonymous

Anonymous Tue 22 Feb 2022 19:24:53 No.14256512 Report

Quoted By: >>14256518

>>14255386
>The sum of infinitely many numbers isn't a number
Delusion

Anonymous

Anonymous Tue 22 Feb 2022 19:26:01 No.14256518 Report

Quoted By: >>14256693

>>14256512
what does that sum equal? -1?

Anonymous

Anonymous Tue 22 Feb 2022 20:15:40 No.14256693 Report

Quoted By: >>14256786

>>14256518
Yes

Anonymous

Anonymous Tue 22 Feb 2022 20:47:27 No.14256786 Report

Quoted By: >>14256872

>>14256693
Speaking of delusion.
2/0=3/0
2=3

Anonymous

Anonymous Tue 22 Feb 2022 20:55:49 No.14256822 Report

Quoted By:

>>14254597
>>14254720
>>>z = -1/(n-1)
>that 1 shouldn't have a negative sign but this seems interesting
It's literally just the geometric series. The closed form is 1/(1-x) at least for x < 1.
>But what about x > 1 1/(1-x) outputs values for that range; for example -1 for x = 2
Yeah, what about it?

Anonymous

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Anonymous Tue 22 Feb 2022 21:04:26 No.14256849 Report

Quoted By:

>>14254294
I honestly fail to see anything fundamentally wrong with this. If you multiply 1+2+4+8+... by 2 and add 1, you get 1+2+4+8+...
x*2+1=x does define a specific number: -1.
What can you do with -1 that you could also do to 1+2+4+8+... and arrive at an inconsistency?

Anonymous

Anonymous Tue 22 Feb 2022 21:09:20 No.14256866 Report

Quoted By: >>14256883 >>14258381 >>14258752

>>14254438
>>14254296
Why?

Anonymous

Anonymous Tue 22 Feb 2022 21:11:45 No.14256872 Report

Quoted By: >>14256885

>>14256786
Dividing by zero is incoherent nonsense.

Anonymous

Anonymous Tue 22 Feb 2022 21:13:22 No.14256883 Report

Quoted By: >>14256898 >>14256954 >>14258358

>>14256866
Because the result is nonsense unless you change your whole concept of what a sum is, such that the "sum" of positive numbers could be negative, disconnecting it from the way addition actually works, and creating a situation where the '+' in your infinite sum just doesn't mean the same thing it means in any other context.

Anonymous

Anonymous Tue 22 Feb 2022 21:13:51 No.14256885 Report

Quoted By: >>14256898

>>14256872
It is not more a nonsense than multiplying by ?.

Anonymous

Anonymous Tue 22 Feb 2022 21:17:49 No.14256898 Report

Quoted By: >>14256907

>>14256883
An infinite sum of rational numbers can be irrational. Infinite sums don't need to have the same properties as finite sums.
>>14256885
Correct. Both are equally nonsensical.

Anonymous

Anonymous Tue 22 Feb 2022 21:22:05 No.14256907 Report

Quoted By: >>14256915

>>14256898
Pointless argument about semantics. If your "sum" somehow gives a negative value for positive summands, your "sum" is clearly something other than what is normally meant by a sum, which is why work with divergent sums sets its own special context.

Anonymous

Anonymous Tue 22 Feb 2022 21:25:50 No.14256915 Report

Quoted By: >>14256938

>>14256907
Pointless argument about semantics. If your "sum" somehow gives a irrational value for rational summands, your "sum" is clearly something other than what is normally meant by a sum.

Anonymous

Anonymous Tue 22 Feb 2022 21:32:24 No.14256930 Report

Quoted By:

>proof by sleight of hand

Anonymous

Anonymous Tue 22 Feb 2022 21:34:22 No.14256938 Report

Quoted By: >>14256944 >>14256957

>>14256915
You're an just autistic, contrarian midwit. There's nothing inherently wrong with generalizing a concept, but in this case, you do so at the cost of departing from the most basic and intuitive properties of a sum, which hold in every other context, except for one marginal case. What do you gain in return? Nothing. Maybe just the satisfaction of feeling "smart" despite having no intellectual accomplishments whatsoever. Eh. Just fuck off; there's a reason why no one agrees with you.

Anonymous

Anonymous Tue 22 Feb 2022 21:36:45 No.14256944 Report

Quoted By: >>14256956

>>14256938
Seethe.

Anonymous

Anonymous Tue 22 Feb 2022 21:41:15 No.14256954 Report

Quoted By: >>14256962

>>14256883
>Because the result is nonsense unless you change your whole concept of what a sum is,
what exactly is the change? where exactly is the error?

Anonymous

Anonymous Tue 22 Feb 2022 21:41:23 No.14256956 Report

Quoted By:

>>14256944
Seethe.

Anonymous

Anonymous Tue 22 Feb 2022 21:42:33 No.14256957 Report

Quoted By: >>14256969

>>14256938
>you do so at the cost of departing from the most basic and intuitive properties of a sum, which hold in every other context, except for one marginal case
And you do the opposite at the expense of dispensing with the distributive property, which is fundamental an axiomatic when reasoning about addition and multiplication. Which isn't something vague like your idea.

Anonymous

Anonymous Tue 22 Feb 2022 21:44:39 No.14256962 Report

Quoted By:

>>14256954
>what exactly is the change?
Read the post.

>where exactly is the error?
There is no real "error" except in abusing the word 'sum' when your result of your manipulations doesn't live up to the properties implied by that word.

Anonymous

Anonymous Tue 22 Feb 2022 21:45:40 No.14256969 Report

Quoted By:

>>14256957
>And you do the opposite at the expense of dispensing with the distributive property
How?

Anonymous

Anonymous Tue 22 Feb 2022 21:45:47 No.14256970 Report

Quoted By:

this thread is just basic calculus concepts that they teach college freshmen
bros, i think /sci/ is larping...

Anonymous

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Anonymous Wed 23 Feb 2022 11:36:32 No.14258273 Report

Quoted By: >>14258306 >>14258323

>tfw we have threads like this daily but my ln2 thread gets axed for shitposting

Anonymous

Anonymous Wed 23 Feb 2022 12:02:28 No.14258306 Report

Quoted By:

>>14258273
>rearranging the terms
You're just breaking the logic of the sequence that way.

Anonymous

Anonymous Wed 23 Feb 2022 12:17:39 No.14258323 Report

Quoted By:

>>14258273
https://en.m.wikipedia.org/wiki/Riemann_series_theorem

You'll learn this in your first semester of studying math.

Anonymous

Anonymous Wed 23 Feb 2022 12:37:31 No.14258358 Report

Quoted By: >>14258388

>>14254296 >>14254438 >>14256883
You can't because you just can't okay.
What books do I need to read for a rigorous understanding of why I can't do this.

Anonymous

Anonymous Wed 23 Feb 2022 12:47:32 No.14258381 Report

Quoted By:

>>14256866
Well if you find an application where it is useful in physics or engineering then you can, or you can do it in a maths journal where you clearly state your foundations you are redefining and explore the implications(I think it's been done)

Anonymous

Anonymous Wed 23 Feb 2022 12:51:16 No.14258388 Report

Quoted By: >>14258398

>>14258358
The last post you link to explains why you "can't" unless you want to change what a 'sum' means. No one said you "can't" assign a value to that thing, though. I'd expect this board to have more high-functioning autists, but you're all low functioning...

Anonymous

Anonymous Wed 23 Feb 2022 12:58:39 No.14258398 Report

Quoted By: >>14258402

>>14258388
>you can't because...because...sum doesn't work like that

Anonymous

Anonymous Wed 23 Feb 2022 13:00:13 No.14258402 Report

Quoted By: >>14258470

>>14258398
>are you saying mathematical concepts have heckin' established meanings?!
Yes.

Anonymous

Anonymous Wed 23 Feb 2022 13:02:43 No.14258409 Report

Quoted By: >>14258411

>>14255336
that's exactly what it does, idiot

Anonymous

Anonymous Wed 23 Feb 2022 13:04:26 No.14258411 Report

Quoted By:

>>14258409
>that's exactly what it does
Explain how it does that without logical nonsequiturs. I'll wait.

Anonymous

Anonymous Wed 23 Feb 2022 13:36:17 No.14258470 Report

Quoted By: >>14258611

>>14258402
>established meanings
Then use propositional calculus

Anonymous

Anonymous Wed 23 Feb 2022 14:26:22 No.14258567 Report

Quoted By: >>14258580

>>14254592
>>14254597
isn't there a theroem that says that you can rearrage diverging sums to obtain any number? pretty sure it was in rudin

Anonymous

Anonymous Wed 23 Feb 2022 14:33:31 No.14258580 Report

Quoted By:

>>14258567
picked it up and you need a convergent series that does not converge absolutely, still pretty cool though

Anonymous

Anonymous Wed 23 Feb 2022 14:50:51 No.14258611 Report

Quoted By:

>>14258470
>use le heckin' propositional calculus
Why?

Anonymous

Anonymous Wed 23 Feb 2022 15:46:23 No.14258752 Report

Quoted By:

>>14256866
because the rules say so

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