Sum of 1 to ∞

What's the sum of 1 to infinity?

I'm not retarded enough to think -1/12

But sums by definition are finite.

Is the question nonsensical?

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en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_⋯
youtube.com/watch?v=sD0NjbwqlYw
en.wikipedia.org/wiki/Cardinal_number
en.wikipedia.org/wiki/Completeness_of_the_real_numbers
wolframalpha.com/input/?i=infinity
myredditnudes.com/
twitter.com/NSFWRedditVideo

>Is the question nonsensical?
Yes because infinity is not a number so there can be no sum of anything with infinity

(1.504785 × 10^186) volume of universe in planck lengths

(2.312437104× 10^62) current age of the universe in planck time

(3.47972×10^248) amount of numbers G it would take to give every planck length in the universe a unique number, for every planck time that has passed since the big bang.

(2.046507 ×10^196) number of years it would take to count every number in G assuming you could count one number every planck time.

(13.7 × 10^9) current age of the universe in years

(10^103) predicted age of the universe when heat death will occur

(2.046 × 10^93) number of consecutive universes from birth to heat death required to finish counting G, which didn't even really matter to count, but also was a bajillion universes ago so having completed counting is completely irrelevent to information in the current universe.

Infinity doesn't need to exist in maths. There is literally no reason a real number would even have to be as large as any number listed here, yet all of the numbers here are infinitely small compared to infinity, even despite the fact that the universe is often considered infinite.
10^93 consecutive universes were required to finish counting G at the fastest speed possible. A billion universes would be 10^9. A trillion would be 10^12. Not a billion, not a trillion, not even a grape ape gorillion. 1 with 93 zeros after it. That many individual full life term universes needed to exist in order to finish counting G, an arbitrary number based on stupidly miniscule but finite values.

You can't even make it two minutes pondering how big infinity could go counting in increments of a million each second.

Infinity doesn't need to exist in math. The concepts of eternity will remain, but a number quantity infinity is just goofy bullshit. It's too big to matter.

Adding finite values to infinity results in infinity ya dumb cunt

>sum by definition are finite
This is not true for indefinite sum, which can converge (ex. 1+1/2+1/4+.....+1/inf=2) or simply diverge like 1+1-1+1.....+1-1...= no finite value. Then you can continue them in weird ways and obtain that 1+2+3+4.....=-1/12 but this is another story

>Infinity doesn't need to exist in math
Neither does logic, then. Math is used as a means of distraction for the intellectual masses who would other overthrow the state were they not preoccupied wasting their life away solving proofs. Imagine thinking this arbitrary nonsense has any basis on real life

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> infinitely

infinitesimally, brainlet

You dont need logic at all. You're right.
It would just be shitty math.

What's your point?

Infinitesimal is a made up word that means nothing. All numbers are equal to 0 compared to infinity.

*tips*

>But sums by definition are finite.
Would the sum of an infinite amount of 1s be finite? Just counting up for eternity wouldn't end up at some number, when you reached any finite number you'd still have to continue counting towards infinity.
The sum you're asking about is just different in that it's adding bigger and bigger numbers as you count further, just taking bigger steps when counting doesn't mean that you get an end to the counting.

[math] \aleph_0 [/math]

>monkey count many banana

if I walk forever, at which point will I be standing when I stop walking? see the fallacy? the sum diverges.

>thinks it's an N
wew lad

Yeah, learn to fucking count. Monkey's can do it better than you. You make it to 10 before assuming infinity.

>monkey tummy full so many banana

>sums by definition are finite
What? No. [math]\lim_{x\to\infty}\sum_{i=1}^ni[/math] does not exist. The infinite sum of all naturals diverges. In very non-rigorous language you could say [math]\sum_{i=1}^{\infty}i=1+2+...=\infty[/math]. The -1/12 meme has to do with analytic continuation and only makes sense to string theorists/Ramanujan.

Fuck. The first limit sould be [math]\lim_{n\to\infty}[/math]

So you admit converging to infinity means nothing?

Is that perhaps because infinity means nothing?

>So you admit converging to infinity means nothing?
>Is that perhaps because infinity means nothing?
You didn't understand a goddamn thing I just wrote, did you? There's an infinite number of infinite sums that converge to a finite number.
[math]\sum_{n=1}^\infty\frac{1}{2^n}=1[/math]

>lim answer isn't a real number
>so lim doesn't exist
[citation needed]

>calls the limit an "answer"
>why can't something that diverges be said to have a limit?
Fucking brainlet

The sum 1/2^n doesn't equal 1, and extrapolating a convergence is meaningless to arbitrary math. You only have a leg to stand on if you admit convergence is rounding and therefore an approximation.

Btw here you go, you waste of air. This is why it doesn't exist.

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>the absolute state of Veeky Forums
en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_⋯
>converges absolutely
At least learn the definitions of the words you're using before speaking.

I don't care what a wikipedia article written by retards has to say on the matter.

[math]0 \rightarrow \infty = \overbrace{\underbrace{0,1,2,3,4,\cdots}_{\infty \text{ elements of } \mathbb{R}}, \underbrace{\infty}_{\text{not in } \mathbb{R}}}^{\text{all possible elements}} \\ \text{Mapped between 0.9 and 1} \\ 0.9 \rightarrow 1 = \overbrace{\underbrace{0.9, 0.99, 0.999, 0.9999, \cdots}_{\infty \space \mathbb{R} \text{ elements of the map}}, \underbrace{1}_{\text{not in the }\mathbb{R}\text{ map}} }^{\text{all possible elements}} [/math]
If there exists a value to bridge the gap between 0.999... and 1 thus allowing 0.999... = 1, there also exists a value to bridge the gap between real numbers and infinity, thus allowing infinity to be equal to a real number.
If there exists no value to bridge the gap between 0.999... and 1 thus assuming 0.999... = 1, there also exists no value to bridge the gap between real numbers and infinity, thus assuming infinity to be equal to a real number.

Because the value does not actually exist and infinity cannot be reached, there is no possible value to add to 0.999... to make it reach 1; it will never reach 1. No amount of increments in the reals will reach infinity, so no mapped amount of increments between 0 and 1 will reach 1.

0.999... is not "infinitely close" to 1. It is actually infinitely far away from 1. Any arithmetic that shows 0.999... = 1 is therefore flawed by making inconsistent and mistaken assumptions about the construction of a repeating decimal extended from a poor interpretation and implementation of infinity, because infinity has classically always been poorly interpreted and implemented.

[math]0.\bar{9} \neq 1 [/math]

Not even 9/10^n equals 1, much less 1/2^n which is clearly a lesser value. Infinity doesn't mean what you think it means.

God I hope this is bait.

Prove it wrong if you can :^)

>uses the word "mapped" incorrectly
>"bridge the gap" ???
>"cannot be reached" ???
And you wonder why nobody takes you seriously.

I don't have to prove it wrong. You're making retarded logical leaps and you either don't know the mathematical definitions of or are misusing many of the words in that "proof" of yours.

>doesn't know how to map values
>doesn't know how to read english
>think's his judgement matters
It's obciously beyond your scope so just take a breather on the sidelines.

>all of mathematics since the 17th century is wrong
>not me tho lol

Admitting you can't prove it wrong but claiming it's wrong was the worst possible response to "prove it wrong"

You asking to prove that wrong is like asking someone to prove the statement "aosdisiudfhoisdofij" wrong. It's nonsense, it CANNOT be proven wrong because it doesn't mean anything. When language is consistently misused, there's absolutely nothing to go off of.
BTW:
1/9=0.111111......
9*1/9=9*0.1111111....=1=0.999999
There ya go. If you bring up "approximations" or "it's only infinitely close," it's because you don't fully understand the rigorous definition of an infinite sum. You're on the internet, you have lots of resources to learn this stuff.

Explain which real number comes directly before infinity then after you can't do that, realize there then exists a missing gap of information between real numbers and infinity.

There's your gap.

And you probe the point you have a bad understanding of repeating decimals. Yes, [math]0.\bar{1} × 9 = 0.\bar{9}[/math], but the mistake you made was assuming [math]\frac{1}{9} = 0.\bar{1}[/math] instead of the obvious fact that [math]\frac{1}{9} > 0.\bar{1}[/math]

Is 1/9 > 0.1? Yes
Is 1/9 > 0.11? Yes
Is 1/9 > 0.111? Yes
Is 1/9 > 0.1111? Yes
Is 1/9 > 0.11111? Yes
Continue infinitely and you should realize 1/9 > 0.111...

>Explain which real number comes directly before infinity then after you can't do that, realize there then exists a missing gap of information between real numbers and infinity.
What? I never claimed infinity to be a real number. No real number "comes before it."

>1/9 > 0.111...
Please then, how do you represent 1/9 in decimal form if 1/9! =/= 0.11111....?

Also, there isn't any "missing information" with infinities. The definition of a limit to infinity has already been posted in this thread (by me, like 5 posts above). I genuinely believe you would enjoy an analysis course, provided you could actually understand the material.

youtube.com/watch?v=sD0NjbwqlYw

There you go, it doesn't converge and the -1/12 interpretation is fucking stupid.

Assuming you meant != or =/=, there is no exact way to display the decimal of a repeating decimal, just the approximation.
You could attach work to the decimal such as [math]\frac{1}{9} = 0.\bar{1}_{\frac{1}{9}}[/math] where arithmetic plays out as [math]\frac{1}{9} × 9 = 0.\bar{1}_{\frac{1}{9}} × 9 = 0.\bar{9}_{\frac{9}{9}} \rightarrow 0.\bar{9}_{\stackrel{\leftarrow}{1}} = 1 [/math], which is the best way I've personally used it. Without this though, the repeating decimal is obviously an approximation because if it were finitely equal to the fraction, it wouldn't be repeating with a looping remainder, would it?

I don't care what a Veeky Forums shitpost written by a retard has to say on the matter.

I know this is bait or you're retarded but whatever. When you pass to limits strict inequalities become weak inequalities i.e. if for all natural numbers [math]n[/math] we have [math]a_n

There is a theorem that says that if a series is convergent then the values it sums tend to zero.
The natural numbers don't tend to zero and so the series must not converge.

Relies on a misinterpretation of infinity, however.

Describe to me the size of the following sets:
A = [0,1,2,3]
B = [4,5,6,7,8,9]
C = [ [math]\mathbb{R}[/math] ]

Sums to finite things are finite. If you do it infinitely, it literally is -1/12 :^)
weight of your mother

1/n ought to converge then :^)

[math] \aleph_1[/math]

en.wikipedia.org/wiki/Cardinal_number

Retard
read the way the implication goes
If series converges then summands tend to zero
not the converse, which would be if summands tend to zero series converges
A theorem is NOT it's converse

>relies on misinterpretation
No it doesn't. It relies on the definition of limits to infinity, which is clearly and rigorously defined.
>describe the size
|A|=4
|B|=6
|C| is undefined

sort of
inf + finite = inf

fuck I missed the
>smiley with a carat nose

>"infinity is rigorously defined, retard"
>infinity is undefined

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>t. someone who's never seen analysis
infinity is not the same as a limit to infinity, fucko

Appropriate, but still relies on infinity to be well defined.

/thread

Limit to 5 is not the same as summing 5 times?

>If there exists a value to bridge the gap between 0.999... and 1
en.wikipedia.org/wiki/Completeness_of_the_real_numbers

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Correct. It's so obvious you have no idea what you're talking about, I dont know why you keep responding.

Limit to 5 literally is summing 5 times though, you absolute retard.

>Limit to 5 literally is summing 5 times though
?????????????

Take the whole post in all at once and prove to me you can internalize completeness.

what did he mean by this

First prove to me that you can describe mathematics accurately. Start by deleting every post you've made in this thread.

It's literally not

Heres how this goes. Infinity is not a finite number. You think you understand that already, but you really don't in application. Because you think infinity is a finite number, you allow that 1/2^n with infinite work achieved (past tense) equals 1 , or that 1/9 = 0.111... has achieved (past tense) an infinite amount of 1's, and because the infinite has been achieved, a full step of final, finite equality has been achieved. Because you believe to increment to infinity as a finite number, you also believe to increment to 1 from 0 if all the integers 0 to infinity were translated decimally between 0 and 1.

But infinity isn't a finite number. You can't have achieved it ever, and final equality by using infinity was never achieved either. 1/2^n is always seperated from 1 by [math]1-2^{-n}[/math] for any real finite n, and similarly there can only ever be some arbitrary finite amount of repetitions in a repeating decimal.

But then you will serrepticiously treat infinity as a number to claim [math]1-2^{- \infty} = 1-0 = 1[/math] and that it were a real number achieavable by incrementing n within the infinite sum while disregarding a zero partial sum has no relevance to the total and is therefore implied the sum of 1 from 1/2^n you're looking for ought to have been completed at n prior to infinity in the reals, yet you already know n at any real doesn't let it sum 1.

I know you know this.

This post is such nonsense I can hardly digest it. What i hoep youre trying to say is infinity only makes sense in the context of limiting processes. In that case, 1/2+1/4+... absolutely converges to 1.

Infinity doesn't make sense at all. Not as as an unreal number, not as a real number, and not as a simple direction to never stop. It makes the most sense as a direction of unending work, but then garbage is extrapolated from that much as thinking sum1/2^n = 1.

Convergence is just a roundabout label for rounding and approximation, all things considered. If it means to imply direct equivalence, then it is explicitly flawed.

What part of the definition of lim_(x to inf) f(x) makes no sense? What part of the definition of convergence makes no sense? I don't know what work means in your context.

Work is the collective partial sums from n to the limit.

A 4k resolution screen has 8294400 pixels.

The possible images of this resolution in only 2 colors is greater than 9.23x10^2408239, which is greater than all the numbers in your post combined (multiplicative).

>:^)

>Because you think infinity is a finite number, you allow that 1/2^n with infinite work achieved (past tense) equals 1
Mmm no.

>infinite work achieved (past tense)
This is not math, try again.

So work is a set? Or is work a limit? Or is work one of the partial sums? You aren't being clear at all. Your math is shit.

Its actually [math]3×10^{13}[/math].

>is it a set?
>IS IT A LIMIT?
>IS IT oNE PARTIAL SUM¿

>Work is the collective partial sums from n to the limit.
the problem isn't my writing, it's your reading.

Darn, lets try it with 7000000 colors which is the possible colors the human eye can see.

1.7x10^1309030, right?

What does that prove?

It took more than one universe just to count a number with a couple hundred 0's. It wasn't about who can make bigger numbers, it was that infinity is infinitely larger than even your number, and your number may as well be infinitely larger than the numbers in the post, and that these numbers are not useful at their size so how could infinity be useful when it is significantly greater than needlessly big?

wolframalpha.com/input/?i=infinity
An unbounded quantity that is greater than every real number.

Thats not a good definition cause it treats infinity as a real number, which it isn't.

Just bored and wonder about things like if the possible movies less than 2 hours at 4k resolution is greater than the possible games of League of Legends less than 2 hours. I need numbers that big to answer my question so they have a use to me.

Other people wonder about other things and require much bigger numbers for their purposes. It seems like you're trying to shame us because the numbers we need to answer our questions are too big.

If that is the case then my response would be stand out of my light.

It specifically says it isn't a real number.

if infinity was a real number, it would be bigger than itself, by definition.
so it isn't a real number, by definition

If infinity weren't a real number, arithmetic performed on it would be undefined

[math] 1 - 2^{- \infty} = [/math] undefined

Like this.

Rather, [math]\frac{\mathbb{R}}{\infty} = 0 [/math] is what is considered valid.

You're right, i am trying to shame you for being retarded. If you end up with a number larger than the amount of planck times in the life cycle of the universe, you have a retarded number that isn't worth anything. You have an arbitrary amount of 4k arrangements of pixels and it doesnt mean anything. Its math that doesnt produce a result with value

you don't seem to understand
When we write infinity it is only to intuitively describe what is happening. Really what we care about is "limiting behavior", what happens when you input an arbitrarily large number into a function or get arbitrarily close to a "limit point". The symbol is just notation

Well then you further don't understand that any arbitrary real n in sum 1/2^n doesn't equate 1, therefore the sum never will equate 1.

you've just made a statement that is not wrong but misses the point to a level that is so extremely high that I can't begin to understand what you don't understand

>But sums by definition are finite.
holy shit

Heres what you don't understand
1/2^n = 1 if and only if there is a maximum real finite number and it is reached.
Without a maximum real finite number, the divisions can become arbitrarily smaller forever and ever. With a maximum real, dividing that number n in [math]\frac{1}{2^n} = 1-2^{-n}[/math] by 2 would create an underflow much the same as $0.01 / 2 doesn't equal $0.005, so the afterlimit is reduced to nothing and you acquire a smallest possible real positive number that can be divided by 2 to equal 0. The underpinning of this all is that the largest possible number to allow these methods must also be a real number. If you cant increment to it, the method no longer works.

You can't increment to infinity so the method doesn't work, first and foremost. Even if you decided it were possible to arbitrarily increment to and achieve infinity, this would further prove that the value you have is a real and finite number, except you have no possible idea to determine what it actually is. It must be real and finite to allow you a half division resulting in 0, even though by definition the underflow on an infinite decimal would never occurr so you get nothing but an arbitrarily long decimal that can't underflow to zero.

lmao I'm so glad that modern mathematics is just following definitions, not some hipster metaphysical shit like whether it's possible to do an infinite number of steps and shit.
just imagine that being a mathematician 300 years ago meant you had to deal with retards like on a regular basis.

Unfortunately for you, you're so retarded that you don't realize the classical implement of infinity literally introduces inaccuracy.

no, That's not what the notation means at all
you can't just substitute into the limit, not even when the function is defined at the point. You can only do that if the function is continuous and the limit point is finite and the function is defined at the limit point.

Limits are not about the value the function takes at that point, there about what happens as you get closer and closer to a point.

The classical way isn't the way we are talking about in any way. We are talking about tending to infinity as a way of saying "past this point this function strictly stays in a box that gets smaller and smaller as I go further and further".

That's the limit definition.

a_n -> L where L is finite as n -> infinity means that the further you go up the natural numbers, the tighter and tighter a box the values of the function will form around L

a_n -> infinity as n -> infinity means that we can choose a large value M past and for this value there is a point in the sequence past which all values in the sequence are strictly larger than this value.

There is no number "infinity" here, notice how I haven't used it outside of the notation at the end of an arrow, it's about how boxes around points and platforms below points and if you truly fail to grasp this you are very very lost.

Yet you can't write equations.

[math]f(x)\rightarrow L[/math] as [math]x \rightarrow p [/math]
and [math]\lim_{x\rightarrow p} = L[/math]
have exactly the same meaning, both are standard in the mathematical literature.
The first one is generally better notation though as writing lim gets people into the nasty habit of presuming limits exist before they have shown that.

[math]\lim_{x\to p}f(x) = L[/math]
even

Okay, so what was M

This is the first time I am reading about this and your guys' arguments are fascinating. Here is what I think:
It is really the number of significant digits that define how "large" a number is. In essence, mathematics is a system to quantify things. Any count could be theoretically transformed to form crazy huge numbers, or alternately, compact small numbers. Like 99 and 10^9. Both those numbers contain approx the same amount of information in quantitative terms. The number 101248573 is quantitatively bigger than either of those, because it contains much more information.

觀察一個由十二意見分

四反駁由三個事實分 = -4/3

>基於七項天上戒律

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“一維寬度字形”和“偏移實體寬度”?

You might be completely fuckin braindead, my guy. It's been spelled out to you over and over in this thread and you're still not grasping it.

地理解。 中國人喜歡數學,因為它可以幫助他們確定他們希望使用的字符; 字母筆劃順序加筆劃數是文化歷史。

Understood. Chinese love mathematics because it helps them identify the character they wish to use; letter stroke order plus stroke count is cultural history.

Linguistic parsing milestone reached.

簡體通用結構

Braindeath = ? ||| 腦死亡=導電虛空

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