Convergent series

[Anonymous No.9452822]

[Anonymous No.9452841]

>>9452822
>>>/lit/

[Anonymous No.9452846]

youre mangling notation really badly

[Anonymous No.9452848]

>>9452822
hey OP what is the limit of this sequence? If you need to, feel free to look up the definition of a limit.
.9, .99, .999, .9999, .99999, ...

[Anonymous No.9452885]

>>9452848
classical limits are unattainable

[Anonymous No.9452894]

>>9452885

[Anonymous No.9452896]

fixed

[Anonymous No.9452897]

>>9452822
Are you fucking serious dude

[Anonymous No.9452903]

>>9452822

[Anonymous No.9453113]

нeт

[Anonymous No.9453125]

>>9453113
based

[Anonymous No.9453127]

>>9453113
But they are at the same point, there's no distance between them.

[Anonymous No.9453139]

>>9453127
no of course

[Anonymous No.9453149]

>>9453127
even though the distance between the two points is infinitely small, they are still fundamentally separate

[Anonymous No.9453151]

>>9453149
But they're not.

[Anonymous No.9453179]

>>9452822
Congratulations, you misunderstand convergence. Slap yourself for being retarded and then go back to Calc 2.

[Anonymous No.9453197]

Reads like something from Professor John Gabriel's chef-d'œuvre
https://drive.google.com/file/d/0B-mOEooW03iLd3gyMGlWNk02amM/view

[Anonymous No.9453200]

>>9452822
>not using the cauchy sequence method of proving 0.9 = 1

[Anonymous No.9453384]

>>9453197
eople from Wikipedia do not agree with you and with him.

[Anonymous No.9453411]

here u go OP
[math]
.99999... = \lim_{n\to\infty} \sum_{i=1}^{n} \frac{9}{10^{i}} = 1
[/math]

[Anonymous No.9453426]

>>9452896
Thank you

[Anonymous No.9453434]

>>9452822
It's the time of the year when all the calc 2 students are learning seq/series and keep making threads about them

[Anonymous No.9453445]

>>9453149
They are representations, there is no distance between them in the first place. Distance is an illusion caused by belief in a number line. Don't fall for dogmas children.

[Anonymous No.9453477]

>>9453411
false

[Anonymous No.9453482]

>>9453477
thats literally the definition of the decimal expansion of a real number

https://en.wikipedia.org/wiki/Decimal_representation

[Anonymous No.9453486]

I propose to officially exclude John Gabriel from the scientific community. He is engaged in pseudoscience.

[Anonymous No.9453521]

>>9453384
People from wikipedia don't have PhDs in vector spaces

[Anonymous No.9453596]

>>9453486
I support your decision.

[Anonymous No.9453600]

>>9453521
People from Wikipedia are smarter than pseudo-scientist-loser.

[Anonymous No.9453610]

>>9453521
Source?

[Anonymous No.9453637]

>>9452885
... for you.

[Anonymous No.9453661]

John Gabriel is not allowed to wear the title of professor.

[Anonymous No.9453665]

>>9453637
Read the definition of the limits.

[Anonymous No.9454933]

>>9452822
1 = 0,999... + 0,000... ...001

[Anonymous No.9454938]

>>9454933
Oh so NOW we can count infinite zeroes?

[Anonymous No.9454944]

>>9454938
it's an infinity between two known points

[Anonymous No.9454962]

>>9452822
This n = 0.000...001 is false.

Suppose n = 0.000...001.
n + 2 ends in an odd digit.
[math]\sqrt{n+2} ends in an odd digit \implies \sqrt{2} ends in an even digit. [\math]
Clearly contradictory to math done over two thousand years ago and math done today.
n = 0.000...001 does not end in an odd digit.
n is undefined.

This is just one of many contradictory results which will always arise when inconsistent assumptions are taken.

[Anonymous No.9454967]

>>9454944
This n = 0.000...001 is false.

Suppose n = 0.000...001.
n + 2 ends in an odd digit.
[math]\sqrt{n+2} ends in an odd digit \implies \sqrt{2} ends in an even digit. [/math]
Clearly contradictory to math done over two thousand years ago and math done today.
n = 0.000...001 does not end in an odd digit.
n is undefined.

This is just one of many contradictory results which will always arise when inconsistent assumptions are taken.

[Anonymous No.9454974]

>>9454967
I don't get this logic t b h

[Anonymous No.9454993]

>>9454967
Please enlighten me, what is the last digit of sqrt(2)?

[Anonymous No.9455013]

>>9452822
Two numbers are NOT the same if there is SOME finite difference between them.
Please show this finite difference.
...
Can not be done, therefore they are the same number.

[Anonymous No.9455016]

>>9455013
see >>9454933

[Anonymous No.9455022]

>>9455016
FINITE number. That is not a finite number.

[Anonymous No.9455026]

>>9455022
finite = what has an end

[Anonymous No.9455028]

>>9454993
It is odd and even whatever it happens to be. A better why of looking at it is to say it does not terminate.

[Anonymous No.9455029]

>>9452896
No you didn't, there are still =/= signs where there should be = signs.

[Anonymous No.9455035]

>>9453149
Saying that .9 repeating is not the same as 1 is as retarded as claiming that .5 and 1/2 are different numbers because "HUR DUR 5 IS NOT THA SAM AS 1 AND 2 HAHA DUR I SO SMRT!!!!1!ONE!!111!!"

.9 repeating and 1 are different representations of the same number. There is no reasonable argument to the contrary. You are wrong. Get over it. Accept that math is not your calling in life.

[Anonymous No.9455038]

>>9453521
Neither do you if you think 1 =/= .999...

[Anonymous No.9455039]

>>9455026
finite does NOT equal what has an end!
one third is a finite number with an infinite number of digits needed to represent it.

[Anonymous No.9455042]

>>9454933
Wrong. The ellipses at the end of the 9 means that there are an INFINITE number of nines. That 0.000...001 retardation means that there are a FINITE number of zeros between the decimal point and 1.
You fail, go back to whatever shithole in Europe you came from.

[Anonymous No.9455046]

>>9454944
That's not what was fucking written. It's not infinite just because you're dumb enough to think it is.

[Anonymous No.9455053]

>>9455046
>>9455042
what's the smallest possible number

[Anonymous No.9455054]

>>9452822
>>9452896
0.9+0.09+... denotes a LIMIT
It's MEANINGLESS otherwise. You can't add an infinite infinite amount of numbers.
This means, that by definition 0.9+0.09+...= limit(partial sums) = 1

[Anonymous No.9455067]

>>9455054
>You can't add an infinite infinite amount of numbers.
wrong

[Anonymous No.9455073]

>>9455054
>>9455067
Also, let me add into it.
You can not only add an infinite amount of numbers into this, but as you are only adding nines into it, you know for sure that the end of it is 9
So, just imagine you are "fast-forwarding", warping, into the end of this number, you can only see nines.
so 0,999... 999.

[Anonymous No.9455104]

if we just swapped to using base .999— instead of base 10 we wouldn’t have this issue

[Anonymous No.9455135]

>>9455073
>>9455073
Again, you can't know for sure it is a 9. That is still an assumption.

Suppose:
[math] n=\lim_{x\to\infty} \dfrac{10^x-1}{10^x}=0.99...9\\
m=\lim_{x\to\infty} \dfrac{10^x-2}{10^x}=0.99...8\\
s=\lim_{x\to\infty} \dfrac{n-m}{10^x}=0.00...1\\
[/math]
s is undefined because it defines the terminating digit of an irrational number as even or odd. m or n(or both) is an inconsistent definition.

[Anonymous No.9455144]

>>9455135
Correction:
s = n - m = 0.00...1
(Although in either instance s is the same)

[Anonymous No.9455146]

>>9455135
your supposition is exactly what I'm saying it is
is it that impossible? why can't it ever be like this?

[Anonymous No.9455149]

>>9455146
My idea is that it is inconsistent or contradictory. Not sure if that is what you were saying or not.

[Anonymous No.9455164]

>>9455149
I'm saying the very basic axioms are wrong on this and 0,000...1 is a real number. Also 0,999...9 is a number.

[Anonymous No.9455171]

>>9453482
this

[Anonymous No.9455176]

fuck all you stupids

.9 repeating = 1

.000...001 doesn't fucking exist, it isn't even well defined in the hyper reals

.3 repeating = 1

and yes, = equals, not some gay fucking arrow

[Anonymous No.9455195]

>>9455164
If you chose to define the number that way, then we have to say
0.333...3 = 1/3
3(1/3) = 1 = 0.999...9
1-0.999...9 = 0.000...1 = 0.000...0

Technically, under this definition having n+0.000...0 OR n+0.000...1 is the next largest number after n. The additive identity element, "0", becomes very muddy in this choice.

[Anonymous No.9455205]

>>9455195
but it's not 0,333...3
it's something bigger than that, but smaller than an ending 4.
it can't be represented like that, but it can by saying 1/3

[Anonymous No.9455231]

>>9455205
You would be defining a different set of numbers using different axioms.

[Anonymous No.9455250]

>>9455231
a different set than the current in use or the one we are talking? I don't see why it can't be like that, it's just that some numbers cannot be represented, 0,999... not being one of them

[Anonymous No.9455328]

>>9455250
there don't exist Real numbers that differ at an infinitely far away decimal. All numbers can be represented since the Real numbers are complete

https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

[Anonymous No.9455364]

>>9455205
yeah, I'm believing Wolfram|Alpha over a random 4chan shitposter
https://www.wolframalpha.com/input/?i=(0.333...)+-1%2F3

[Anonymous No.9455443]

okay, faggots, if you think you are so smart, and think those are two different real number, there must be numbers between them
give me a decimal representation of any number you want between 0.9999.... and 1.000...
1.999....5 is not a valid candidate because you can't do that in real numbers, that is, say there's an infinite sequence of numbers which somehow ends in a different number
0.9999..... is not 0.999....9, because the sequence doesn't end
fuck off

[Anonymous No.9455525]

>>9455035
Are you ready to be wrong?

1-epsilon equals .9 repeating


Therefore 1 cant be .9 repeating or epsilon is 0. It isnt.

[Anonymous No.9455545]

>>9455053
Reals are Archimedean, so there are no infinitely small not infinitely big numbers, that is numbers smaller or bigger than any other real number

[Anonymous No.9455551]

>>9453477
>false

[Anonymous No.9455566]

>>9453665
limits are not defined as "attainable"

[Anonymous No.9455585]

>>9455545
is 0,000... = 0?

[Anonymous No.9455614]

>>9454933
0,000... ...001 = 0

[Anonymous No.9456200]

>>9455038
Maybe you should look up "Bilinear forms"

[Anonymous No.9456216]

Y'all fucking brainlets. Clearly OP is working with the non-Hausdorff space [math]\mathbb{R}\cup\{\overline{1}\}[/math].

[Anonymous No.9456459]

>>9456216
you mean [math]\mathbb{R}\cup\{\overline{.999...}\}[/math]?

[Anonymous No.9457819]

https://www.youtube.com/watch?v=2TCDiK7GpNM
geometric series -- best video on youtube ever (apart from the on e.) I'm not even kidding.

[Anonymous No.9457834]

Convergence absolutely does not have the same value or result as equation and anyone who writes an infinite sum is "equal" to something should be shot in the back of the head when they least expect it.

Also
>>9444882

[Anonymous No.9457840]

>>9452822
The DEFINITION of equality is the convergence you show...

[Anonymous No.9457844]

>>9457834
>Convergence absolutely does not have the same value or result as equation
Convergence is the DEFINITION of EQUALITY.

[Anonymous No.9458088]

>>9457844
No.

[Anonymous No.9458099]

>>9458088
Uneducated retard.
Go learn something, you clearly have no clue what you are talking about.

Have a look when two real numbers are equal.

SPOILER: Since a real number is just the equivalence class of cauchy sequences of rational numbers, they are equal iff they are in the same cauchy series, by definitions that is iff their difference converges to zero.

[Anonymous No.9458396]

>>9453521
>PhDs in vector space
That's a good one lul

[Anonymous No.9458466]

>>9458396
he's bas(is)ed

[Anonymous No.9458535]

>>9458099
You are legitimately retarded if you couldn't follow the math presented in >>9444882

[Anonymous No.9458627]

Fun fact: Infinity is never used anywhere in calculus
[math] \forall \epsilon>0,\ \exists M>0:n>M \implies\ |(\sum^n_{k=1}\frac{9}{10^k})-1|<\epsilon [/math]
prove me wrong

[Anonymous No.9459498]

>>9455525
Epsilon is 0, 1 - .(9) = 0.(0) =0

[Anonymous No.9459507]

>>9455551
that filename lmao

[Anonymous No.9459513]

>>9458627
moron
https://en.wikipedia.org/wiki/Laplace_transform#Formal_definition

[Anonymous No.9460649]

>>9452822
We were told that the = regarding a limit has its own definition. The real value it approaches OR a improper limit (uneigentlicher Grenzwert), I guess undefined could be the result too, if there is a 1/0 or something in there?

[Anonymous No.9460653]

>>9452822
The line about 0.9999... being unequal to 1 is kind of wrong, because the limit sum is basically included in that.

[Anonymous No.9461406]

>>9459513
[eqn]F(s)=\int^\infty_0f(t)e^{-st}dt=\lim_{a \rightarrow \infty }\int^a_0f(t)e^{-st}dt \iff \\ \forall \epsilon>0,\ \exists M>0:a>M \implies \bigg |\int^a_0f(t)e^{-st}dt-F(s) \bigg |<\epsilon[/eqn]

[Anonymous No.9461690]

>>9460653
1 = (9/10 + 1/10) = 0.9 + 10/100
= 0.9 + (9/100 + 1/100) = 0.99 + 10/1000
= 0.99 + (9/1000 + 1/1000) = 0.999 + 10/10000
= 0.999 + (9/10000 + 1/10000) = 0.9999 + 10/100000

and so on

each line = 1
exactly one, not approaching it.

line #1 is exactly 1
line #10 is exactly 1
line #98327498236483689 is exactly 1

At infinity, it still is exactly 1

[Anonymous No.9461710]

>>9455551
>angry concave brainlet.png