Mathematics - 2012-09-11 (page 2 of 3)

1:00 PM

Gah! stupid return, stop sending it!

$(z-16)(z-4)=0\\
(x^2-16)(x^2-4)=0\\
x=4 or 2$

Then we substitute in to get y

$\begin{cases}
xy=8\\
x^2+y^2=20
\end{cases}\\
y=\frac{8}{x}\\
y=\frac{8}{4} or \frac{8}{2}\\
y=2 or 4\\
(4, 2); (2,4)$

Jayesh Badwaik

$x^2 = a^2 \implies x=a$ only?

Dan the Man

yeah.. right?

Jayesh Badwaik

What are the solutions for $x^2 = 9 = 3^2$?

Dan the Man

Oh.... right.....

$3$ and $-3$

Jayesh Badwaik

NEVER FORGET!!

Dan the Man

1:07 PM

Yes, sir!

Nick Kidman

@anon: I should not have waited with my proof of the abc-conjecture to release. Well, whatever, it was a fun afternoon working out the proof.

Jayesh Badwaik

@NickKidman :P :P :P Unless you are serious of course.

Dan the Man

@JayeshBadwaik So where did I go off?

Peter Tamaroff

@nick whaaaaa¿

Jayesh Badwaik

$(x^2-16)(x^2-4)=0\\ x=4 or 2$

Is this complete?

@DantheMan ?

Dan the Man

1:09 PM

Ah

x = 4 or 2 or -4 or -2

Nick Kidman

afk, drawing Feynman diagrams for my paper with the proof of the Yang-Mills mass gap problem

Jonas Teuwen

@NickKidman What. TikZ?

Nick Kidman

Jayesh Badwaik

@DantheMan So you have got the solution right?

Dan the Man

Yeah! $$(4,2)\\
(2,4)\\
(-4,-2)\\
(-2,-4)$$

Jayesh Badwaik

1:12 PM

Good.

@DantheMan Now the addition subtraction part.
You have terms $x^2 + y^2$ and $xy$, what are some popular combinations which you can see of those terms?

Dan the Man

Ah!

Perfect square trinomial

Jayesh Badwaik

@anon In the comment to this answer . Do you idea of what a structure of selfrefferentiality is?

Since MO does not have links to comments.

@DantheMan Can you do something with it? The perfect square thing?

Dan the Man

Yeah

Jayesh Badwaik

try then.

Dan the Man

$\begin{cases}
2xy=16\\
x^2+y^2=20
\end{cases}$

Add them

And get

$x^2+2xy+y^2=36$

$(x+y)^2=36$

$x+y=\pm 6$

John Junior

1:19 PM

12 mins ago, by Jayesh Badwaik

What are the solutions for $x^2 = 9 = 3^2$?

Dan the Man

@JohnJunior Riiiiight :)

John Junior

12 mins ago, by Jayesh Badwaik

NEVER FORGET!!

Dan the Man

Yes

fixed it

John Junior

;-)

Dan the Man

$y=\pm 6 -x$

2 hours ago, by Dan the Man

$\begin{cases}
xy=8\\
x^2+y^2=20
\end{cases}$

$x(\pm 6-x)=8$

$\pm 6x -x^2=8$

Not sure what to do now.

Ah

$-x^2 \pm 6x = 8$

John Junior

1:24 PM

2 hours ago, by John Junior

What are you looking for?

MaoYiyi

any know anything about the university of south africa?

Dan the Man

$-x^2 \pm 6x -8=0$

What is a $\pm$ times $-1$?

@MaoYiyi Not me. sorry

MaoYiyi

@DantheMan thanks, just thinking of where to goto graduate school.

Dan the Man

@MaoYiyi cool

MaoYiyi

@DantheMan any suggestions outside of USA?

Dan the Man

1:30 PM

@MaoYiyi For mathematics?

MaoYiyi

@DantheMan yes.

John Junior

7 mins ago, by Dan the Man

$-x^2 \pm 6x -8=0$

John Senior

H roomies

Dan the Man

@MaoYiyi There is UNAM, in Mexico City. The have a mathematics career. UNAM is ranked 44th best university in the world.

Jayesh Badwaik

@JohnSenior Hola amigos!

John Junior

1:33 PM

@DantheMan This is two equations, so write it as two equations please.

Dan the Man

@JohnJunior Ah ok

MaoYiyi

@DantheMan is mexico city safe?

John Senior

@JayeshBadwaik kak dela?

Dan the Man

$-x^2+6x-8=0\\
-x^2-6x-8=0$

@MaoYiyi Relatively!

@MaoYiyi You have to know when and where not to go, and stuff. But Mexico City is way safer than the rest of Mexico

brb

MaoYiyi

@DantheMan just asking because of the bad things I have seen on the news

Jayesh Badwaik

1:36 PM

@JohnSenior I am fine. Thank you.

John Senior

@JayeshBadwaik :)

Jayesh Badwaik

@JohnSenior :P

I have to rush now, so talk to you later. bye.

Dinner calls.

peoplepower

@JohnSenior Oh no, you're senior again!

John Senior

@peoplepower I have been senior for quite a while now - since I stopped being old a week or two ago :)

John Junior

Stop being old and be senior :)

John Senior

1:40 PM

and changed my gravatar back after it started giving Will nightmares :)

peoplepower

Now, this is awkward, maybe a case of a mental disorder.

Jonas Teuwen

I am a war machine. Roarrr.

John Junior

I am a peace machine. Purrr.

peoplepower

I am a machine. I am a machine. I am a machine ...

Peter Tamaroff

@nick whaaaaa

MaoYiyi

1:44 PM

I am anti-machine!

John Junior

Welcome to the machine.

Dan the Man

@MaoYiyi Ah yea. the media always exaggerate everything

Jonas Teuwen

@JohnSenior There is a What section now!

John Senior

@JonasTeuwen excellent - my lack of understanding is growing

Jonas Teuwen

@JohnSenior That's the goal.

John Senior

1:53 PM

@JonasTeuwen :)

MaoYiyi

if you upvote this will I get points? hehe

Jonas Teuwen

@JohnSenior I added the why too!

John Senior

@JonasTeuwen you are right - my lack of understanding really is growing

Jonas Teuwen

Great.

Nick Kidman

what the heck is this

John Senior

1:56 PM

My lack of understanding of Mochzuki's proof of abc is probably unbounded :)

Jonas Teuwen

@JohnSenior I hope it will grow exponentially.

MaoYiyi

@NickKidman a problem I don't understand.

John Senior

@NickKidman might be a joke - or it might be a genuine result - hard to tell

Nick Kidman

reminds me of https://www.destroyallsoftware.com/talks/wat

Jayesh Badwaik

@NickKidman that is awesome .Enough talking about languages that suck. Lets talk about javascript. LOLZ

John Junior

1:59 PM

25 mins ago, by Dan the Man

$-x^2+6x-8=0\\
-x^2-6x-8=0$

@DantheMan Have you finished?

You need to be interested enough to follow it through to the end.

N3buchadnezzar

Hi =)

anon

@JayeshBadwaik Don't know.

Nick Kidman

How is the Ackermann function not primitive recursive, if every value for n is determined by values involving only numbers lower than n?

anon

2:23 PM

"Every value for an argument is determined by values at lower arguments" - is this equivalent to being primitive recursive?

Nick Kidman

proably not, I have no intuition what the "primitive" means

it's a list of things defining the class of primitive recursive functions, but this is of course not practical in seeing if and why some function is not in it

anon

"The primitive recursive functions are the basic functions and those obtained from the basic functions by applying these operations a finite number of times." -- Wikipedia [see the article for "these operations"]

The number of times $A$ has to be applied to itself becomes arbitrarily large as we increase the arguments, so it doesn't fit the mold.

Also the recursive definition of A does fit the definition of a primitive recursion, except for one detail: the two functions being composed in the recursion would themselves have to be primitive recursive, but we cannot assume this a priori for A(-,-) lest we be circuitous.

Nick Kidman

okay, so in short the -primitive- recursive functions have a feature that the computation time is in some way stronger bounded than general recursive functions

Can someone explain to me how the if-function is implemented in a real life computer?

anon

On what level, assembly language?

Circuits?

(Not that I can answer the question anyway.)

Zhen Lin

Regrettably, I only know it at the level of assembly language...

Jonas Teuwen

2:33 PM

@ZhenLin Not low level enough?

You expand into a tree and select the branch.

Zhen Lin

@NickKidman A complete proof of the non-PR-ness of Ackermann can be found in my notes, Thm 1.4.5.

Jayesh Badwaik

@NickKidman The function is implemented as a multiplexer I would say, if you are asking about hardware.

Jonas Teuwen

The code is expanded linearly, and the previous conditions determine to which spot the counter jumps, so to speak.

Or you want it on transistor level...

THEN!

anon

@ZhenLin There is a citation needed on Wikipedia that might be relevant to you.

Jayesh Badwaik

Multiplexer

In electronics, a multiplexer (or MUX) is a device that selects one of several analog or digital input signals and forwards the selected input into a single line. A multiplexer of 2n inputs has n select lines, which are used to select which input line to send to the output. Multiplexers are mainly used to increase the amount of data that can be sent over the network within a certain amount of time and bandwidth. A multiplexer is also called a data selector. They are used in CCTV, and almost every business that has CCTV fitted, will own one of these. An electronic multiplexer makes i...

Or a demltiplexer, which is just opposite of it.

Jonas Teuwen

2:36 PM

@JayeshBadwaik Thanks Sherlock... that's what you learn in high school, no?

N3buchadnezzar

Hey

Jonas Teuwen

I think I've learnt that in the fourth year.

Parallel -> Serial + clock.

Jayesh Badwaik

@JonasTeuwen Are you making fun of me bro?

Jonas Teuwen

Paradoxically that often allows higher transmission speeds!

@JayeshBadwaik No. 8-).

Zhen Lin

@anon Hm, so it is. But I don't think citing personal notes is acceptable. :p

Jonas Teuwen

2:37 PM

(cross talk and such!)

@ZhenLin Would it be possible to add doi links to our own documents or do we need some "publisher key" for that?

anon

@ZhenLin You could suggest it on a talk page and let others decide for you then.

Jonas Teuwen

If you manage to give a persistent link I guess it is okay.

Jayesh Badwaik

@JonasTeuwen :P
Yup serial is faster than parallel for anything above 100's of megahertz.

Jonas Teuwen

@JayeshBadwaik Hmm, I don't think the speed has that much to do with it as a sharp boundary.

One of the issues is capacitive coupling between the wires.

Also, it messes up the slew-rate.

Jayesh Badwaik

@JonasTeuwen Yup, and in the current copper/silver,gold the capacitive coupling becomes problematically significant in that range.

Zhen Lin

2:39 PM

@JonasTeuwen I'm not sure where we'd get a prefix.

Jonas Teuwen

@JayeshBadwaik Hmm. Isn't it about... the distance and insulator between the conductors and not such much about the conductors itself?

Path length also.

Nick Kidman

@ZhenLin: k, thanks

Jonas Teuwen

@ZhenLin But would be cool.

Maybe MSE can get one, so you can link to posts!

Nick Kidman

And regarding the if-question, I'm interested in the hardware, yes

like if you would be stranded in 1940, could you tell them what to do to build a computer?

Jonas Teuwen

Yes.

Jayesh Badwaik

2:41 PM

@NickKidman then multiplexer.

Jonas Teuwen

First make gates, bro.

You only have tubes.

Jayesh Badwaik

@JonasTeuwen And those (tubes) are huge!!

Jonas Teuwen

Pretty damn huge indeed.

Zhen Lin

Tubes are basically mechanical switches. They managed to build computers with them, after all.

Jonas Teuwen

There is such a thing as a multiplexer tube apparently...

Crazy glass blowers.

The first transistor looks like a modern piece of art. That's quite disturbing.

Jayesh Badwaik

2:43 PM

It looks like a haphazardly grown bonsai.

Mariano Suárez-Alvarez

@wj32 You should review the definition of maximal... :-)

Jonas Teuwen

Something that if you eat it you will die a horrible painful death.

With all the sharp corners and all. Punctures the intestine.

Nick Kidman

Are all computable functions "just" recursive?

Zhen Lin

By definition.

Nick Kidman

so do all computable function have a fancy register machine picture I could hang up as poster on my bedroom wall?

Jayesh Badwaik

2:49 PM

@NickKidman That is kind of point of Turing Machine and Church-Turing Thesis.

Zhen Lin

In principle. Or an even fancier Turing machine.

Nick Kidman

I.e. the starting tape together with the what-to-do-grid

I just wonder, because the script I'm reading is only talking about integers atm.

Jonas Teuwen

According to Zielberger, that's nonsense. No such thing as infinite!

Nick Kidman

like sin(x) with x=0.5333..., is this to be approximated by a code using only integers in the register?

Zhen Lin

Yes.

Real numbers, contrary to the name, do not exist in general. :p

Jonas Teuwen

2:52 PM

Henceforth they are actually imaginary...?

Creepy!

Nick Kidman

they are, of course :)

Jonas Teuwen

A product of our minds.

Nick Kidman

there has actually been a wild fight on the physics board about the reality of imaginary numbers

Mariano Suárez-Alvarez

no there has not

Nick Kidman

6

Q: QM without complex numbers

I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?

Mariano Suárez-Alvarez

2:54 PM

sigh

Jonas Teuwen

Hahahahaha.

I am almost crying of laughter.

Also, let us kick out Banach spaces.

Nothing is infinite dimensional in real life!

Zhen Lin

pffft

nothing is infinite in real life!

Jonas Teuwen

They should meet up with D. Z.

Are there also physicists trying to "find the physical meaning" of abstract intermediate quantities?

"is an electron an object?"

After that, they've put the philosophy department in another building.

Nick Kidman

there are sure threads on this..

user19161

Well, the truth is, nobody knows whether space and time are infinite, except possibly one man.

Jonas Teuwen

2:59 PM

Quite some arcane physicists.

Hmm, they should strictly split metaphysics and physics when discussing it. Me think. But me retard.

Well, I mean... make sure you know which is which.

user19161

The argument that there is a creator of the universe because time must have a beginning is nonsense.

Nick Kidman

the thing is, eighter you ask the physical question you have purely mathematical, i.e. "in general relativity, is there..." or you ask them about the real world "I'f I'm out in space and I..." and of course you get philosophical very easy, as people will explain things about the real world with the (mathematical) theory they've learned

user19161

@NickKidman either

Jonas Teuwen

@NickKidman The answer is neither yes nor no, henceforth the question is wrong.

Zhen Lin

Assuming the universe doesn't have strange geometry, the observable universe has finite volume, so it may as well be finite.

user19161

3:01 PM

@JonasTeuwen You like to use henceforth bro? I am not sure that is the correct usage.

Jonas Teuwen

@WillHunting Henceforth I need to tinkle.

@ZhenLin What if it is non-measurable... Oh no.

Then we can split it in two equal ones.

user19161

Poincare's conjecture aka Perelman's theorem essentially narrows down the possibilities of the shape of the universe.

Jonas Teuwen

I don't dare to think about the consequences, let me get a beer.

user19161

More beer, less spacetime.

Zhen Lin

Doubtful. I have yet to see convincing evidence that the universe is a simply-connected compact smooth 3-manifold.

Nick Kidman

3:03 PM

I also find it fascinating how different mathematicans have a different understanding of how mathematics relates to reality. I read the article by I think Arnold some months ago, which basically started with "Mathematics is a part of physics."

Doing geometry feels like doing physics sometimes

N3buchadnezzar

=)

Zhen Lin

I'm sure there are physicists who say physics like doing geometry...

user19161

Well, physics motivates much of mathematics, and mathematics is used to solve many physical problems. End of story.

Jonas Teuwen

It does not relate at all. We just think it does. Like the way physics relates to reality. It is all in our heads... phew. Beer!

Basically what we are doing is rather desperate and henceforth nonsense.

Or the other way around.

user19161

I just asked my friend to help me get the two Lee manifolds books so I don't need to pay for shipping. Phew!

Nick Kidman

3:06 PM

I also find it interesting how "computer science" bacame a subject on its own

they only learn some special math topics

Mariano Suárez-Alvarez

well, computers are sort of important these days...

user19161

Well, theoretical computer science intersects with much of mathematical logic.

Nick Kidman

I've never used a computer in my life

user19161

I use computers mostly to do naughty things.

Mariano Suárez-Alvarez

we can only wish...

user19161

3:08 PM

It is rare for Mariano to chat so much...

Zhen Lin

The parts of theoretical computer science not concerned with explicit bounds may be construed as mathematical logic, but complexity theory has the feel of combinatorics, really...

Nick Kidman

"Look at them, with their snobby glasses, sitting at Starbucks with their Turing machines, trying to look edgy."

So they talk a lot about computability of the characteristic function of a predicate here. But computation of it aside, isn't it difficult to construct the function in the first place?

Zhen Lin

If you have defined it well enough to talk about it, then you have constructed it. What more is there to it?

user19161

@jacky Just in case you did not know, you can accept an answer and upvote at the same time. :-) — Will Hunting 22 secs ago

user19161

Hehe, I am so desperate for points.

user19161

3:11 PM

@nick You were talking about your proof of abc, is that a joke or for real?

Nick Kidman

I'm a physicists and 25 yo.

user19161

So?

Mariano Suárez-Alvarez

computer science is much more than explicit bounds and mathematical logic... from the design and implementation of programming languages to database theory to lots of other stuff

Nick Kidman

So no, I basically have no idea about prime numbers

Jonas Teuwen

@MarianoSuárez-Alvarez That's applied CS! A real computer scientist doesn't use a computer.

Mariano Suárez-Alvarez

3:12 PM

you have never met anyone who works on implementing functional languages...

:-)

user19161

Oh dear this jackyboi is using this site to check all answers to his homework it seems...

Mariano Suárez-Alvarez

they have the analogue of grothendieck primes

Zhen Lin

A seminar speaker once said, "I'm a theoretical computer scientist, so for me a programming language is an extension of the simply-typed $\lambda$-calculus."

and on he went with the seminar...

user19161

I know lambda and calculus but not lambda calculus.

Nick Kidman

who is talking with whom here?

Jonas Teuwen

3:14 PM

@NickKidman TO ME.

user19161

@NickKidman Chaos in action.

Mariano Suárez-Alvarez

this is an onanist room

we all talk to ourselves

Zhen Lin

@MarianoSuárez-Alvarez I have difficulty imagining a TCS version of a Grothendieck prime.

user19161

@MarianoSuárez-Alvarez I happen not to understand that word. I have a tiny vocab. :-)

Mariano Suárez-Alvarez

yet the very first adjective you can come up with is tiny...

user19161

3:15 PM

Yes, I only know about 18,000 words according to testyourvocab.com

user19161

The test takes only 5 min.

Nick Kidman

3:27 PM

afk

user19161

3:38 PM

Oh man, this jacky accepted my 2 answers but did not upvote? I won't answer him again!

Mariano Suárez-Alvarez

just have fun and forget about votes

user19161

Yes, but votes is part of the fun. :-)

Mariano Suárez-Alvarez

@NickKidman, (there is no need to inform anyone that you are afk)

user19161

@gustavo Are you around?

Jayesh Badwaik

@JonasTeuwen the only real computer scientist I know who does not use computer is donald knuth

and I know what your response will be "You mean, there are others of that type????"

:P

John Senior

3:47 PM

@WillHunting I saw a guy who accepted an answer and then asked for loads of explanation ... and finally posted his own answer (in much better English than the question)

user19161

@gustavo I have something that will help you. See tutorial.math.lamar.edu for excellent notes on algebra, calculus, linear algebra and differential equations. Would help you prepare for your university course.

Gustavo Bandeira

@WillHunting I'm here

@WillHunting Oh, thanks for it. =)

user19161

@GustavoBandeira Those notes are fairly complete. I think they are exactly what you need at this stage.

Gustavo Bandeira

Yep.

I'll use them with the OCW lectures.

user19161

@GustavoBandeira I think you may not need the OCW or the books you got. :-) But you just do what is best for you.

Gustavo Bandeira

4:00 PM

@WillHunting Yep, it seems paul notes are pretty complete.

user19161

4:17 PM

@GustavoBandeira Of course, I would not recommend you shitty things.

Gustavo Bandeira

@WillHunting =)

user19161

4:33 PM

Hey @rob! I was aiming for low hanging fruit again...

robjohn

@WillHunting where?

user19161

@robjohn I answered 7 questions today. :-)

robjohn

@WillHunting wow, that is a bunch.

unNaturhal

4:55 PM

Good morning folks

Peter Tamaroff

palindromes!!!

unNaturhal

Could someone help me with series? :/

Peter Tamaroff

@unNaturhal Yes.

@unNaturhal But he's away.

unNaturhal

@PeterTamaroff Oh..

Peter Tamaroff

@unNaturhal Instead of someone, I can help. Would that be good?

unNaturhal

5:00 PM

@PeterTamaroff ...? Are you kidding me? o.o

Peter Tamaroff

@unNaturhal I can.

@GustavoBandeira

@unNaturhal So what's the question?

unNaturhal

@PeterTamaroff Eh eh.. I haven't understood nothing about series..

I started to study it today

Peter Tamaroff

@unNaturhal Well, let's start with a simple one.

What is a series (of numbers)?

Or better.

What is a number sequence?

unNaturhal

But the only think that I have understood is that a serie $a_n$ is a sum of terms

Peter Tamaroff

@unNaturhal OK, we can do as follows.

We know a sequence is a function $f:\mathbb N\to \mathbb R$, that is, to each natural number $n$, it assigns a real number $a_n$, yes?

So we can talk about the sequence $\{a_n\}_{n\in \mathbb N}$

in general terms.

Correct?

unNaturhal

5:05 PM

@PeterTamaroff Yes

@PeterTamaroff Yes, correct

Peter Tamaroff

@unNaturhal OK.

So, we now define a new sequence in terms of the other, as follows:

$$s_1=a_1$$

$$s_n=s_{n-1}+a_{n}$$

So we get (check it yourself!)

$s_2=s_1+a_2=a_1+a_2$

$s_3=s_2+a_3=a_1+a_2+a_3$

unNaturhal

Yes

Peter Tamaroff

And we can see, in general that $s_n=a_1+\cdots +a_n$

We usually write this as $$s_n=\sum_{k=1}^n a_k$$

So, for some sequences $a_n$, it is meaningful to assing a value to $\lim \;s_n$

And that is the idea of series.

However

series needn't be infinte.

unNaturhal

Mh mh...

Ok and.. How I can obtain $s_n$?

Peter Tamaroff

We obviously get finite series with finite sequences, and infinite series with infinte sequences.

@unNaturhal Well, that depends on the $a_n$

But what is troubling you?

unNaturhal

5:12 PM

Because on my book there is some formulas like this:
$$s_n = \sum_{k=1}^n{\frac{1}{k(k+1)}}=\frac{n}{n+1}$$

Peter Tamaroff

@unNaturhal Well, let's see.

Can you express $$\frac 1 {k(k+1)}$$ in partial fraction expansion?

unNaturhal

Yes, it is:
$$\frac{1}{k}-\frac{1}{k+1}$$ right?

Peter Tamaroff

Yep

@unNaturhal Now, consider $s_n$ again.

$$\sum\limits_{k = 1}^n {\left( {\frac{1}{k} - \frac{1}{{k + 1}}} \right)} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 1}^n {\frac{1}{{k + 1}}} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 2}^{n + 1} {\frac{1}{k}} $$

unNaturhal

@PeterTamaroff Oh...

@PeterTamaroff And how this can help us?

Peter Tamaroff

@unNaturhal Think for a moment what the last two sums are. In particular $$\sum\limits_{k = 2}^{n + 1} {\frac{1}{k}} = \sum\limits_{k = 1}^n {\frac{1}{k}} + \frac{1}{{n + 1}} - 1$$

unNaturhal

5:19 PM

@PeterTamaroff Mmmh... maybe there are some proprieties of $\sum$ that I dont know..

Peter Tamaroff

@unNaturhal Well, it is but only a good symbol to use.

In plain english:$$\sum\limits_{k = 1}^n {\frac{1}{k}} - \sum\limits_{k = 2}^{n + 1} {\frac{1}{k}} = \left( {1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}} \right) - \left( {\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} + \frac{1}{{n + 1}}} \right)$$

Do you see what is going on?

unNaturhal

@PeterTamaroff Maybe.. we can say that the both $\frac1n$ can be cancelled

@PeterTamaroff Oh no, wait

Peter Tamaroff

@unNaturhal Only those?

What about $1/2$?

unNaturhal

All terms, untill the $\frac{1}{n}$ can be cancelled.. so it will be only $\frac{1}{n+1}$

Peter Tamaroff

@unNaturhal What about $1$?

unNaturhal

5:23 PM

@PeterTamaroff Oh, is not common.. so the result should be $$1 - \frac{1}{n+1}$$

Peter Tamaroff

@unNaturhal Purrrrrrfect.

unNaturhal

:D

Peter Tamaroff

@unNaturhal So, all in all you get $$\sum\limits_{k = 1}^n {\frac{1}{{k\left( {k + 1} \right)}}} = 1 - \frac{1}{{n + 1}}$$

So it is but only natural to say $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\frac{1}{{k\left( {k + 1} \right)}}} = \sum\limits_{k = 1}^\infty {\frac{1}{{k\left( {k + 1} \right)}}} = 1 - \mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}} = 1$$

Could we close this troll-question??

unNaturhal

@PeterTamaroff Yeah :)

Peter Tamaroff

@MichaelGreinecker

@robjohn

unNaturhal

5:27 PM

Now I know how to obtain $s_n$ :) I try to understand Cauchy criteria, if I can't I will back :)

Thanks @PeterTamaroff!

Peter Tamaroff

@unNaturhal Anytime.