Simply Beautiful Art's realm of calculus and analysis

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Nilknarf

00:01

Hey @simply!

Simply Beautiful Art

Hey

I mean

I'm not here

>.>

Nilknarf

In my calc class we're doing differential equations

Simply Beautiful Art

Mmm

Separating those differentials all the time

Nilknarf

and now I know what you meant when you said that they can easily be turned into difference equation problems

Simply Beautiful Art

Also

the differential equations too easy

x'D

Nilknarf

00:03

I have a problem for you

...that is, if you aren't busy

Simply Beautiful Art

I'm considering writing more ordinals and pinging my 'students'.

@Nilknarf Eh...

Hit me anyways

Nilknarf

Okay

Simply Beautiful Art

I'll dream it over.

Nilknarf

If $f(x)=x^{\ln(x)+2}$, find $f^{\circ n}(x)$.

Simply Beautiful Art

Mmm....

Okay

Does it come out pretty?

Nilknarf

00:04

Oh yes

Simply Beautiful Art

Nilknarf

very satisfying :D

Simply Beautiful Art

Cuz I can always brute-force the solution out.

Nilknarf

haha

Simply Beautiful Art

Nilknarf

00:05

meanwhile, I'm struggling with an infinite series problem :P

Simply Beautiful Art

I haven't been on MSE much lately

Nilknarf

no?

why not?

Simply Beautiful Art

Nah. Its mostly that when I get on it, I end up spending multiple hours...

And I'm kinda busy atm

Also diverting lots of brain power towards my program.

Nilknarf

Ah, I see

code golfing, you mean?

Simply Beautiful Art

Yeah. Once I'm not so busy I'll be back on mse

@Nilknarf yes, a bit.

The codegolf is mostly something I can fit between things

But its hard being on mse for only a few minutes

Nilknarf

00:08

WHOAH

Simply Beautiful Art

fail

Nilknarf

whoops, nevermind

Simply Beautiful Art

lol

Nilknarf

I solved an interesting iteration problem the other day

Simply Beautiful Art

(In secret, I kinda hope that if @user202729 and @Mr.Xcoder learn about the ordinal approximations of TREE(3) well enough, they'll be able to write some programs for the TREE(3) challenge. Even if their solutions aren't super golfy, there's a level of respect I feel for anyone who can even write a program to go up to TREE(3), and for them to have made it that far because of me, there's also a small bit of pride.)

@Nilknarf ofc you did.

Nilknarf

00:13

Haha, ok

those are your "students"?

Simply Beautiful Art

I guess

Need to also check in with the calculus people

Nilknarf

ARGH, help me

Simply Beautiful Art

@Nilknarf ?

Nilknarf

$$\sum_{n=1}^\infty \arctan\bigg(\frac{(-1)^{n+1}}{F_{n+1}(F_n+F_{n+2})}\bigg)$$

D:<

I know it's gonna telescope, but I can't figure out how!

Simply Beautiful Art

lmao

$F_{n+1} = F_{n+2}-F_n$

Then: drive.google.com/file/d/0BxKdOVsjsuEwdjBEM1dpRkhMa2s/view

Page 3.

and page 4.

Nilknarf

00:17

I know, that's where I got the problem

Simply Beautiful Art

Oh, lol

Well

Split it over even and odd cases.

Nilknarf

jeez, how long is it gonna take for me to get an intuition about this type of problem?!

Simply Beautiful Art

Its supposed to take a while

the problems in that PDF are supposed to be horrendously tricky.

Courtesy of Jack D'Aurizio lol.

Nilknarf

yeah

Simply Beautiful Art

$$\arctan(x)\pm\arctan(y) = \arctan\left( \frac{x\pm y}{1\mp xy} \right)$$

Nilknarf

00:21

right right right, I got that

I already set up a functional equation for my telescope, but I can't solve it :P

Simply Beautiful Art

We could attempt to set $x\pm y/1\mp xy=1/(F_{n+2}^2-F_n^2)$

@Nilknarf oh?

Nilknarf

Yeah

Simply Beautiful Art

Hm...

Nilknarf

$$b_{n+1}=\frac{1-b_nF_{n+1}(F_n+F_{n+2})}{b_n+F_{n+1}(F_n+F_{n+2})}$$

Simply Beautiful Art

oh, no thanks.

Nilknarf

00:23

Heheh, yeah

Simply Beautiful Art

No, bad @Nilknarf

Nilknarf

Simply Beautiful Art

Well

We know that $x(n+2) = y(n)$. Or at least I'll assume such.

Nilknarf

Sure

Simply Beautiful Art

Or not.

Something about that $F_{n+2}^2-F_n^2$ bothers me.

Nilknarf

00:27

aha

Simply Beautiful Art

$F_{n+1}(F_n+F_{n+2})=F_{2n+2}$

Nilknarf

No, that won't help

Hold on, I may have something good

Simply Beautiful Art

It doesn't?

Nilknarf

Okay

if we let
$$x=\frac{\sqrt 5}{2}+F_n^2-F_{n+2}^2$$

Then $$\frac{2x}{1+x^2}=\frac{1}{F_{n+2}^2-F_n^2}$$

Simply Beautiful Art

Nice find

Except

Nilknarf

00:30

and so our sum is equal to

Simply Beautiful Art

Doesn't fit the equation quite right

Nilknarf

$$\sum_{n=1}^\infty (-1)^{n+1}\arctan \frac{2x}{1+x^2}=2\sum_{n=1}^\infty (-1)^{n+1}\arctan x$$

now the real question is:

does that even matter? :P

Simply Beautiful Art

$F_{n}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}$?

Nilknarf

Is that true?

Oh ho! Is that so? XD

Simply Beautiful Art

I mean

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , ...

And hi @user202729

Nilknarf

00:34

Ok, maybe that helps

Simply Beautiful Art

x'D

Nilknarf

argh argh argh

5 hours later…