Mathematics - 2014-09-16 (page 1 of 4)

Chris's sis

00:10

Soon I'll also compute $$\sum_{n=1}^{\infty} (-1)^{n+1}\frac{H_{2n+1} H_{2n+2}}{(2n+1)(2n+2)}$$

Actually I'm curious to see if I can get rid of the hypergeometric function or any polylogarithm function.

I'll post it to myself as "Compute without pen and paper: $t_1=1$ and $t_n=n(1+t_{n-1})$. Find $\displaystyle \prod_{n=1}^\infty \left(1+\frac{1}{t_n}\right)$" and go to sleep. This one is very enjoyable.

Semiclassical

@Winther) An observation regarding your comment to that 'Prime Root Matrix' question: googling "prime root matrix" yields 14 results, and googling "prime root matrix" -"alwyn scott turner" (the author of that question) eliminates all of the the relevant ones. so I think it's pretty fair to characterize that as a crackpot question

Chris's sis

OK, OUT FOR NOW

Random Variable

02:27

@r9m Are you feeling better?

r9m

04:49

@RandomVariable yes ! :) Thanks for asking :-)

Random Variable

04:59

@r9m I have a bit of a cold. Or it might just be allergies. It's hard to say at the moment.

r9m

@RandomVariable oh ! take care then ! There is a outbreak of cough and cold in our hostel ! a lot of guys are getting it .. I got it from someone else .. and now my roommate has got it frm me :( terrible thing

Antonio Vargas

@Semiclassical, if you hover over the downvote arrow you may decide to vote after all ;)

Semiclassical

@AntonioVargas: eh, I'll stick with my initial reaction for now

@AntonioVargas: I'm much more liberal with up-votes than down-votes in general, and i'd rather not react out of pique

Antonio Vargas

@Semiclassical Understandable.

Semiclassical

@AntonioVargas: Though I really do hope someone offers an answer in the spirit of Winther's comment to that question---theta functions were my initial reaction upon seeing that question, but i don't know much about approximating them or their antiderivatives

Antonio Vargas

05:15

@Semiclassical Totally, me neither. O.L. gave an awesome answer on this question of mine using them. They're far outside of my day-to-day tools though so I haven't really gotten a chance to play around with them.

Random Variable

@r9m There was only one time in my life when I got really sick from a cold/flu virus. And that's when I developed pneumonia. I missed 2 weeks of school.

Semiclassical

@AntonioVargas: That is a really slick answer. I know elliptic integrals far more than elliptic functions / theta functions, which is frustrating since I know the latter is very useful in computing elliptic integrals; i just can never quite keep track of it well enough

@AntonioVargas for example

Antonio Vargas

@Semiclassical Wow, that's nuts. It's like the answer would just fall out of your head onto the paper if you know enough!

Along with some other things, perhaps.

Lucas Zanella

05:31

How to write that vertical bar when I want to say that the antiderivative goes from a to b, for example?

Antonio Vargas

@LucasZanella, try \left. stuff \right|_a^b

Lucas Zanella

@AntonioVargas I tried but the bar is too litle

tiny

sorry

Antonio Vargas

$$\left. e^{2 x^2} \right|_0^1$$

Lucas Zanella

Antonio Vargas

Oh I see

Lucas Zanella

05:33

The bar gets tiny

Antonio Vargas

Maybe $$-\cot t - t \Bigr|_{\arcsin(1/3)}^{\arcsin(2/3)}$$

how's that?

Lucas Zanella

Exactly! Thank you so much @AntonioVargas

Antonio Vargas

@LucasZanella No problem.

usukidoll

06:32

@AlexanderGruber do you know pdes?

Alexander Gruber

@usukidoll Ehhh... not really but try me

(this will depend entirely on what level of PDE we're talking about)

usukidoll

@Alexander I am stuck... how does the $\frac{-2}{5} $ go away?
http://math.stackexchange.com/questions/931921/find-the-particular-solution-of-u-x2u-y-4u-exy-satisfying-the-following-s

Alexander Gruber

@usukidoll ok lemme read through

usukidoll

k

Mike Miller

heya @Alex

Alexander Gruber

06:45

hi

usukidoll

@Alex save me D:

Mike Miller

how's things

usukidoll

dying in pde land :) sarcasm

Alexander Gruber

@MikeMiller they're alright. working like a madman

Mike Miller

that sounds like a good state

Alexander Gruber

06:53

it ain't bad.

usukidoll

@AntonioVargas do you know pdes

Antonio Vargas

@usukidoll Not really. I've taken a couple classes on them though. Why do you ask?

usukidoll

I am super stuck... like how does the $\frac{-2}{5}$ disappear when if I am iwrt z I have to get that fraction otherwise my integration would be wrong. http://math.stackexchange.com/questions/931921/find-the-particular-solution-of-u-x2u-y-4u-exy-satisfying-the-following-s

@AntonioVargas

Antonio Vargas

@usukidoll Sorry, it's way too late for me to read all of that. I don't think I can help :)

usukidoll

:( you suck

Antonio Vargas

07:07

How kind.

usukidoll

there needs to be more pde people..then I wouldn't be saying those words

Antonio Vargas

Uh huh.

Studentmath

08:26

Do you have an example of some solved PDE question avaible for you, @usu?

usukidoll

yeah but it had a different side condition @studentmath

and the damn book doesn't go into detail that much

@studentmath do you know pdes

Will Hunting

I don't know any DE, so I can't help you.

usukidoll

^ not u

Will Hunting

I know, I am just saying.

But I intend to learn about them some time in the future.

usukidoll

good luck

it is hell

Will Hunting

08:29

Currently, I know how to solve separable variables, first order linear ODE and second order linear ODE homogeneous with constant coefficients, lol.

usukidoll

dat is ez]

Will Hunting

That is all I need to know to read my differential geometry book.

Hi @robjohn you should be sleeping.

usukidoll

@robjohn do you know pdes?

Will Hunting

@usukidoll Which area of math are you most interested in?

usukidoll

odes r my fav

Will Hunting

08:43

@usukidoll You should read Gerald Teschl's book then. It is the best ODE book I have come across.

usukidoll

I had a crummy ode book , yet I still passed with an A-

Will Hunting

I never did ODE in uni, only in high school, lol.

Hippalectryon

10:30

@Chris'ssis shhhh don't tell ;)

Chris's sis

Greetings

@Hippalectryon can you do Alitzer's problem without pen and paper?

$t_1=1$ and $t_n=n(1+t_{n-1})$. Find $\displaystyle \prod_{n=1}^\infty \left(1+\frac{1}{t_n}\right)$

Will Hunting

10:50

@Chris'ssis Have you latexed your book already?

Chris's sis

@WillHunting No, not yet.

MickLH

11:00

Are you tired of Dubstep that sounds... well... a few iterations short of a workable numerical solution?

Maybe you just need to try something analytic instead ;)
https://soundcloud.com/full-synthetic/inured-feat-humming-bird-militia

Studentmath

11:22

@usu once I am back from the trip with the dog, if you still need help, I will go over my books again and see if I can help

Or if you could give the example problem and the solution, it will certainly remind me, and maybe also get you to understand the new problem better

Balarka Sen

12:00

@DanielFischer I'm looking for rationals in $[\log(n), \log(n+1)]$ with given denominator.

Any suggestions?

Daniel Fischer

@BalarkaSen With given denominator, say $d$, you can look at $\lfloor d\log (n+1)\rfloor$ and $\lceil d\log n\rceil$.

Balarka Sen

yes.

but how does that supposed to help?

Daniel Fischer

Any integer between these two, provided the latter is not smaller than the former, gives a rational in that interval.

Balarka Sen

uh, $n$ and $d$ are not known actually. i am trying for some diophantine results, to clear up what i mean.

Daniel Fischer

$\lceil d\log n\rceil \leqslant n \leqslant \lfloor d\log (n+1)\rfloor \implies \log n \leqslant \frac{n}{d}\leqslant \log (n+1)$.

Okay, so not a given denominator.

Continued fraction?

Balarka Sen

12:06

maybe. i don't know of any good diophantine approximation results on logs

Daniel Fischer

Me neither. What is it that you actually want?

Balarka Sen

well, if you want to know, the problem we (me and a friend of mine) are looking at is to find growth results on $c(n)$, the minimal integer such that there is an integer $x$ such that $2^x$ sits in between $(n+1)^{c(n)}$ and $n^{c(n)}$. all we have is that $c(n) \ll n/2$ (vinogradov notation).

now i am interested in looking at the $x$s. how many could be there for a given $c = c(n)$?

Daniel Fischer

You mean "for how many $n$ does a given $c$ work"?

Balarka Sen

"how many integers could be there in $[c\log_2(n+1), c\log_2(n)]$"

@DanielFischer that problem is already solved.

there are, of course, infinitely many such $n$s, and we have a closed form

yep

it's not hard to invert $c(n) = c$, btw, for large $n$s

Daniel Fischer

There can be roughly $\left\lceil \frac{c}{n\log 2}\right\rceil$ integers in such an interval. More typical, I guess, would be the floor, and while $n$ is not much larger than $c$, you may need to take higher order terms of the logarithm into account.

Balarka Sen

12:23

funny

oh by the way our bound was $c(n) \ll n \log(2)/2$, not $\ll n/2$

Chris's sis

@robjohn here is my work to that alternating harmonic series

Balarka Sen

@Chris'ssis It's not hard to do by generating functions, btw

oh i think you have showed me that before

right right

no polylogarithms

right

Chris's sis

@BalarkaSen Yeah, that was my point.

Balarka Sen

is there a high-school method to evaluate it though?

Chris's sis

@BalarkaSen Yeah. I get the expansion of $\log(\cos(x))$ by elementary tools.

Balarka Sen

12:32

fourier analysis is not high-school!

Chris's sis

@BalarkaSen No Fourier ... I can get that expansion by high school tools.

Balarka Sen

ah. post it here.

Chris's sis

@BalarkaSen Do you want me to teach you a very brilliant way?

Balarka Sen

sure why not

rehband

Is there an easy way of seeing the series expansion of $\log(1+x+x^2)$ ?

Chris's sis

12:38

:17697254

Balarka Sen

@rehband taylor for $\log(1+t)$?

Chris's sis

@BalarkaSen use the result above derived by telescoping sums.

rehband

@BalarkaSen I know that $\log(1+t) = t - t/2 + t/3 - ...$

Balarka Sen

oh noes, i am not going to try your problem. too hard for me, @Chris'ssis

Chris's sis

@BalarkaSen That allows you to get the expansion of $\log(\cos(x))$ (elementarily)...

rehband

12:40

What would be the easiest/quickest way of showing $\lim_{x\to0} \frac{ \log(1+x+x^2) - \log(1+x)}{x^2} = 1$ ?

Balarka Sen

i don't see how but i'll take your word, @Chris'ssis

Chris's sis

@BalarkaSen It's very easy, I let you think of it.

Balarka Sen

i will not think of it. too hard for me =P

Chris's sis

:D

Balarka Sen

@rehband partial factorization and then L'Hopital

or a few terms of taylor would do

rehband

12:44

@BalarkaSen Okay thanks

Balarka Sen

$\log(1+x+x^2) = x - x^2/x + O(x^3)$. $\log(1+x+x^2) - log(1+x) = x^2 + O(x^3)$

$$\frac{\log(1+x+x^2) - \log(1+x)}{x^2} = \frac{x^2 + O(x^3)}{x^2} = 1 + O(x)$$.

As $x \to 0$, the error tends $0$

so the answer's 1

so actually no L'Hopital needed

rehband

@BalarkaSen Hmm, $x^2/x$ ?

Balarka Sen

x^2/x?

oh

i mean $x^2/2$

rehband

Right

But isn't it still wrong? :P

Balarka Sen

why?

oh darn

$\log(1+x+x^2) = x + x^2/2 + O(x^3)$

rehband

12:49

Yes

Now I agree :)

Balarka Sen

anyways that's what i had in mind so my calculations aren't wrong

rehband

Tyty

Balarka Sen

@rehband to calculate taylor of $\log(1+x+x^2)$ : $$\log \left ( \frac{1-x^3}{1-x}\right ) = \log(1-x^3) - \log(1-x)$$

rehband

Is it okay if I write something similar to what you said in an answer on MSE?

Balarka Sen

it took me some time, but was elementary

@rehband sure

rehband

12:51

@BalarkaSen Thanks!

Chris's sis

@BalarkaSen Have you seen the problem I received from Alitzer?

$t_1=1$ and $t_n=n(1+t_{n-1})$. Find $\displaystyle \prod_{n=1}^\infty \left(1+\frac{1}{t_n}\right)$

Balarka Sen

@Chris'ssis interesting, but too hard for me

Chris's sis

@BalarkaSen It's very easy, honestly.

Balarka Sen

i decided that calculus is too hard for me, so i won't think about it much

@Chris'ssis that's nice

it's too hard for me though

Chris's sis

@BalarkaSen You'll have a lot of fun playing with it. Just give it a try.

Balarka Sen

12:58

nah

i won't, trust me

i will stick to my algebra.

rehband

@Chris'ssis Interesting

Chris's sis

@rehband Yeah. Try to finish it without pen and paper.

rehband

$\prod_{n=1}^\infty \left(1+\frac{1}{t_n}\right) = \prod_{n=1}^\infty \left(\frac{t_{n+1}}{nt_n}\right)$

Take logs?

Chris's sis

@rehband no

Huy

@rehband: $t_{n+1} = (n+1) (1 + t_n) \neq n(1+t_n)$.

rehband

13:04

@Huy True :P

Chris's sis

$$\prod_{n=1}^N \left(\frac{t_{n+1}}{(n+1)t_n}\right)=\frac{t_{N+1}}{(N+1)!}$$

Then divide $$t_n=n(1+t_{n-1})$$ by $n!$. The rest is a baby job.

2

Balarka Sen

@anon

anon

mmhmm

Balarka Sen

that was fast O_O

rehband

@Chris'ssis :-O

Balarka Sen

13:09

@anon can we use abc-type arguments to find a certain $k_0$ such that there are only finitely many $k$-tuple with elements in $[c\log_2(n), c\log_2(n+1))$ for some $k > k_0$

anon

that's too difficult for me while I'm doing my combinatorics take-home last minute, unfortunately

Balarka Sen

@anon aha, no problem. have fun with the test.

Chris's sis

@rehband $$\underbrace{\frac{t_n}{n!}}_{\displaystyle a_n}-\underbrace{\frac{t_{n-1}}{(n-1)!}}_{\displaystyle a_{n-1}}=\frac{1}{(n-1)!}$$ and then sum over $n=2$ to $N+1$

Balarka Sen

even though none is listening, corrections : $k$-tuple consists of integers, for any $k > k_0$

and $c$ is given

rehband

@Chris'ssis Ah okay, i see

Chris's sis

13:15

@rehband I'll add this to my book.

Will Hunting

@Chris'ssis You should add everything you can think of to your book.

rehband

@Chris'ssis Weeeee :) The day you release your book will be like xmas for me

Chris's sis

@WillHunting But there is a lot of stuff ... I need to make a selection :-)

Will Hunting

@Chris'ssis Hehe, why not make it a multivolume work? =)

Chris's sis

@rehband lol :-)))

Balarka Sen

13:17

gah we need a number theory room in MO. all they've got is some homotopy room which seemingly consists of super-nerds and their jibberish formulas.

Chris's sis

@WillHunting I might leave the world of mathematics after that ...

(I saw many things that I don't like)

Balarka Sen

@Chris'ssis what do you mean?

Chris's sis

@BalarkaSen Hard to summarize in a few words. I prefer not to talk about it more, not now.

Will Hunting

@Chris'ssis Just like Grigori Perelman.

Balarka Sen

well, Perelman is incomparable

Will Hunting

13:21

Dude, Chris's Sis might be more brilliant than Perelman.

Balarka Sen

he solved a legendary open problem, man!

Will Hunting

So what? It's silly to even compare two people.

Balarka Sen

i am not comparing talents. i am comparing works =P

Will Hunting

Same thing.

Balarka Sen

not really

ok, less trivia time

rehband

13:24

How do I add like a star next to something which I wanna refer to later?

in MSE editor?

Huy

@BalarkaSen: I'm sure you know a lot about free vector spaces?

Balarka Sen

heh, @Huy? what makes you think that?

Huy

No idea, just a wild guess, because you're so smart.

Balarka Sen

you've got a wrong idea about me =P

Huy

Too bad.

Chris's sis

13:26

For instance, to have a paper with the most simple solution ever to the Au-Yeung series rejected it seems to me pretty crazy ... I wouldn't be surprised to see the proof there appears in some book written by I don't know what author.

Balarka Sen

@Huy =P

you'll find my knowledge limited in number theory and abstract algebra

Huy

I thought free vector spaces were part of abstract algebra.

Balarka Sen

i think mostly of ring/field theory while i say abstract algebra =P

it's not that i don't even know what a free vector space is, i just don't know "a lot"

Huy

@BalarkaSen: I'm looking at different definitions of the tensor product because I need those for my work. And one of them defines it as the quotient space of the free vector space of the cartesian product w.r.t. some subspace.

Do you know anything about that definition?

Balarka Sen

i forgot my modules, so no.

i am afraid not

Will Hunting

13:31

I only remember algebra-precalculus now.

Ice Boy

I only remember arithmetic lol

Will Hunting

@Chris'ssis One wonders what kind of article they don't reject.

rehband

@Chris'ssis You are better than that, those people aren't fit to judge you. I find what you do extremely inspiring!

Chris's sis

@rehband Thank you.

@WillHunting Yeah, maybe.

Balarka Sen

Oh @Huy, found an excellent solution of that $x^y + y^x > 1$ ineq

Bernoulli's inequality tells us that $(1+x/y)^{1/x} > 1 + 1/y$, i.e., $\frac{x+y}{x} > 1/y^x$, i.e., $\frac{x}{x+y} < y^x$. Similarly, $\frac{y}{x+y} < x^y$. Add up and QED.

Fun isn't it? Someone on the forum came up with it.

Moron

13:44

@BalarkaSen I think there is a typo.

Balarka Sen

er, yeah right

i meant $\frac{x+y}{y} > 1/y^x$

and $\frac{x}{x+y} < y^x$

Moron

Never mind, the idea was clear. I was just nitpicking.

Balarka Sen

no, no, that's fine. it's good to be nitpicky =P

i need to leave bu-byes

Khallil

14:34

Hey, @Chantry.

Hey, @Ice.

Hey, everyone!

Ice Boy

Hi pal @Khallil

Chantry Cargill

Hi @Khallil :)

Khallil

What have y'all been getting up to today?

Chantry Cargill

Just woke up. I'm supposed to be meeting someone to work on a project in -5 minutes. Just getting the energy to leave the door.

Khallil

You should've left 5 minutes ago, @Chantry?

=P

Chantry Cargill

14:38

Yep lol

To be fair, he didn't exactly give me short notice.

Khallil

Are you going to be solving the Riemann hypothesis, @Chantry?

Chantry Cargill

much*

Maybe if I get a half hour after lunch.

Khallil

=P

Chantry Cargill

Anyway, off to class. Enjoy your day Khallil.

robjohn

@Chris'ssis I haven't looked at that series yet, so I will look at your solution later.

Khallil

14:48

You too, @Chantry. ^_^

Ice Boy

@robjohn do you agree with this?

I don't think I've "read" an entire book on mathematics period
(with the possible exception of the princeton companion)
I prefer to think of all of the books I have access to as one giant book that I can dip into whenever I want

it seems a bit arbitrary to me to divide mathematics up into books

Chris's sis

@robjohn OK

robjohn

@IceBoy I've read sections of math books, but I don't think I've read one cover to cover. I usually know what is in sections that I haven't read and if I want to know more about those topics, I will go back and use them as references. I would rather get the ideas and work out the details myself than read the details from the book.

Will Hunting

@robjohn I will try to read my 12 holy books from cover to cover next year!

robjohn

@WillHunting I wouldn't think you would do anything else :-)

Will Hunting

14:55

It's weird to ask this in math room, but any of you can recommend me a good english grammar book?

Ice Boy

CGEL

Will Hunting

OK, too thick for me. I am looking at something under 500 pages.

robjohn

@WillHunting I don't think I've looked at an English grammar book for over 35 years.

Will Hunting

amazon.com clearly has a preference of american over british dictionaries, lol

Hippalectryon

15:32

@Chris'ssis Heya

Khallil

The internet is a great resource, @Will.

Whenever I'm unsure, I just search it on Google.

Will Hunting

@Khallil Real men read real books.

Khallil

The internet is the greatest book in history, @Will.

Will Hunting

@Khallil It is also full of shit.

Khallil

One man's trash is another man's treasure, @Will.

Ice Boy

15:37

but turning trash into treasure consumes a lot more energy than is needed

Hippalectryon

@Chris'ssis Lol that problem is obvious isn't it ?

Is it well known that $t_n=\lfloor e*n!-1\rfloor$

rehband

@Hippalectryon Obvious is the most dangerous word in mathematics

Hippalectryon

@rehband I thought it was trivial :)

rehband

@Hippalectryon Trivial is the second most dangerous word in mathematics

Ice Boy

third please?

Will Hunting

15:40

Easy is the third most.

rehband

:D

Are u talking about that infinite product, hippo?

Hippalectryon

Yep

rehband

Kk

Ice Boy

it is easy to show that this is trivially obvious

Khallil

Will Hunting

15:47

That user is posting stupid comments again, lol.

Ice Boy

don't read what you find stupid

Will Hunting

I think she does that so that the other user will upvote her.

I think she is extremely crafty.

Ice Boy

it is only someone on the internet

Hippalectryon

@Khallil who ?

Will Hunting

16:00

Finally I have 700 points. 300 more to reach 1000.

Khallil

Is it possible to hand MSE points over to other people?

I'd like to donate all of my points to Jasper.

Will Hunting

You can award a bounty to his answer, I think. Nah, don't do that.

Khallil

What's the difference between $\varnothing$ and $\emptyset$?

There has to be a reason why the two exist separately, without a \var prefix.

(Like $\phi$ and $\varphi$)

rehband

@Khallil Nice, I was wondering how to make that second phi

Anastasiya-Romanova

Does anyone here know why Mr. Mhenni is suspended?

Will Hunting

16:09

@Anastasiya-Romanova No.

Khallil

Wait a second. @Anastasiya, did you change your username recently?

I feel so confused. It's as if I've seen that avatar on someone else's profile.

Will Hunting

@Khallil V-Moy

Khallil

I think it was Valentina's.

YES! Suspicions confirmed!

Anastasiya-Romanova

@Khallil Yes. I change it but you can easily notice that I'm V-Moy

Mr. @robjohn, may I know why Mr. Mhenni's account is suspended? What kind of rule violations he did so that I can avoid it?

Khallil

Have you ever seen The O.C, @Anastasiya?

Will Hunting

16:17

@Anastasiya-Romanova It is confidential.

Anastasiya-Romanova

@Khallil What is O.C?

@WillHunting At least he can give me a slight explanation

Khallil

It's The O.C. but don't worry. I can tell you haven't heard of it, @Anastasiya.

Shinpai shinaide.

Anastasiya-Romanova

@Khallil Yes, what does that mean? Explain please...

Hippalectryon

@Anastasiya-Romanova Hi there

Anastasiya-Romanova

@Hippalectryon Hi too

Hippalectryon

16:22

math.stackexchange.com/a/933837/150347 Rep seekers, bad answers

@robjohn Could you add in your $\LaTeX$ in chat link a Katex renderer ? It's usually faster

Khallil

It looks terrible, @Hippa.

Hippalectryon

@Khallil ?

$\int_0^1\iint_0^1\iiint_0^1\iiint_0^1\int_0^1$

Uh

Balarka Sen

Heh KaTeX

Hippalectryon

@robjohn forget that

Balarka Sen

Nobody beats LaTeX

@Hippalectryon RH?

Hippalectryon

16:33

What's RH ?

Rampant Hippalectryon ?

Balarka Sen

Can't even spell his own name correctly

sneer

Hippalectryon

Hypalaictréon

Balarka Sen

@Hippalectryon Reputation Hysteria

Many users in MSE are affected and infected by RH.

Hippalectryon

ams.org/notices/201409/rnoti-p1024.pdf

Chris's sis

@Hippalectryon So, you introduced some values of the sequence in OEIS and realized that there is an alternative way? My way is very easy. :-)

Balarka Sen

16:36

Heh good catch @Chris'ssis

hahah

Hippalectryon

@Chris'ssis Forget what I said, I realized it didn't lead to any solution

(does it ?)

Chris's sis

@Hippalectryon Take a look at my suggestion in the right panel. :D

Hippalectryon

No -__-

Nevar

I gotta find

Chris's sis

lolll OK :-)

Balarka Sen

you gotta get a spellcheck, @Hippa

Hippalectryon

16:38

@BalarkaSen A spellwhat ? Mai englich is véri gud

Balarka Sen

you need to eeemprove your eeenglish, frenchman

Hippalectryon

@Chris'ssis While I'm searching, have you ever worked on derivating integers ?

user176201

anyone here familiar with the Jacobian in integration?

Hippalectryon

I'm looking for great results one can get from int's derivatives

Balarka Sen

@Hippalectryon derivating integers?

what's that supposed to mean?

Chris's sis

16:42

@Hippalectryon What is that?

Hippalectryon

Uh well basically

Chris's sis

@Anastasiya-Romanova Did Cleo answer my question you asked? (let me guess - No)

Hippalectryon

We define $D(n)$ for $n\in\mathbb{N}$ by:

$D(n)=1$ if n is prime

Balarka Sen

ah, right

Hippalectryon

$D(1)=0$

Balarka Sen

16:44

$D(uv) = uD(v) + vD(u)$

Hippalectryon

Yep

That's the three rules

Balarka Sen

integral derivative, heard of that

Hippalectryon

rjlipton.wordpress.com/2014/08/19/the-derivative-of-a-number

Balarka Sen

you just ringed the bell

Hippalectryon

I found them while reading that article

Balarka Sen

16:44

I found them in OEIS =P

Anastasiya-Romanova

@Chris'ssis As you can see, she didn't give a **** for a kid like me, lol

Balarka Sen

tsk tsk language, kid

Anastasiya-Romanova

@BalarkaSen Censored! I learn from 9Gag

Will Hunting

@Anastasiya-Romanova You mean she did not give a fuck? I see.

Balarka Sen

cool article, @Hippa

Anastasiya-Romanova

16:46

@WillHunting No comment...

Balarka Sen

although I guess @Chris'ssis isn't interested. it's just number theory ={

wait, wait. i know Edward Barbeau!

he wrote a paper on the SCESS identity.

search "Barbeau + Seraj + Sum of cubes equal to square of sum"

Anastasiya-Romanova

Okay guys, I have to sleep now since there's nothing special today on MSE except Mr. Mhenni suspension.

Wait, before I sleep. @Chris'ssis could you please help me to derive $\dfrac{\ln(1-x)}{1+x}$ or $\dfrac{\ln(1+x)}{1-x}$ into sum of series involving harmonic number?

I asked you because you're good at this subject

Chris's sis

@Anastasiya-Romanova What do you think? I can?

Anastasiya-Romanova

@Chris'ssis I don't know, I have no idea. I only know that $\displaystyle-\frac{\ln(1-x)}{1-x}=\sum_{k=1}^\infty H_kz^k$ and $\displaystyle\frac{\ln(1+x)}{1+x}=\sum_{k=1}^\infty(-1)^{k+1}H_kz^k$

Chris's sis

@Anastasiya-Romanova $$\int_0^1 x^{n-1} \log(1-x) \ dx =-\frac{H_n}{n}$$ and $$\int_0^1 x^{2n-1} \log(1+x) \ dx =\frac{H_{2n}-H_{n}}{2n}$$

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$Mathematics$ Mathematics