Why are these called closed forms: pi, e, sqrt(2), ln(3)
Why aren't integrals called closed forms?
What's the point of finding closed forms of integrals or sums?
>>7710383
It actually characterizes the functions into certain groups or categories depending on whether you can find elementary antiderivatives for example. It's a very strong theoretical study, at first sight it's obviously dumb just like "straightedge-compass constructions" seems arbitrary and dumb until you use field theory to study it
>>7710383
>Why are these called closed forms: pi, e, sqrt(2), ln(3)
Closed form is a hazy definition to begin with in my experience. Generally it refers to elementary functions like Sin[x], Sqrt[x], Exp[x], Log[x] or constants like Pi. I can't tell you why such functions are considered elementary. For example, the error function has a relation to Exp[x] but it is considered special.
>Why aren't integrals called closed forms?
Not all integrals can be defined in terms of elementary functions or known special functions without creating new functions altogether.
>What's the point of finding closed forms of integrals or sums?
Because it is easier to describe integrals and sums in terms of what is known out there. Look at your pic OP. It's easier to describe the sum in terms of 3 constants (two Pis and an integer) than it is to wield around an infinite sum.
>>7711042
yeah, closed is sort of a social definition. functions or constructions are considered 'closed' when there is a concensus that they are easy to work with or considered familiar enough to us.
>>7710383
How does one find n if computers can't calculate pi in its totality?
Ho
>>7711085
No but seriously seems impossible.
>>7711085
>>7711086
n here is not a number itself, is a sumation, so n goes from 1 to infinity so actually what you have there is...
1/1 + 1/4 + 1/9 +1/16 +1/25 ....... and that infinite sumation is equal to pi^2 / 6
its not possible to calculate it with a computer although you can just keep iterating and you'll keep getting closer and closer but never there...
>>7711085
the same way people calculated it before computers existed
>>7710383
I'm pretty sure the sum in your picture doesn't converge.
Euler's solution to the basel problem (Its the question of finding the closed form for the series you have in your pic) brought him immediate fame. Why? Because no other mathemetician could do it, thats why.
I dont think he was intentionally trying to solve the problem though, he probably came across it when fucking around. If a sum is not known in closed form, Its fucking impossible to be like "My next goal is to find such and such closed form for integral/sum". NOPE. Not like fermats last theorem where wiles had a window through Taniyama-Shimura and worked torwards it for seven years.
>>7711594
>>7711585
Then educate me. You are useless otherwise. My 3 points still stand.
>https://en.m.wikipedia.org/wiki/Closed-form_expression
>>7711604
jth partial sum is j/n^2, no limit as j goes to infinity.
>>7711630
You are aboslutely right :D
There seems to be an error in OP's pic
>>7711042
>It's easier to describe the sum in terms of 3 constants (two Pis and an integer) than it is to wield around an infinite sum.
But pi^2/6 is just shorthand for an infinite sum too.
If you asked me to choose between calculating digits of pi^2/6 or calculating digits of the sum of inverse squares by hand, do you believe that calculating pi^2/6 is easier? I'll race you.
>>7711776
Its a short hand, and moreover it shows a beautiful connection that isn't obvious between circles and series. How are some people this autistic.
Most people would shorten further to [eqn]\zeta(2)[/eqn]
>>7711788
How do you relate that sum to the circle?
What is the geometric connection of the circle to the Basel problem?
>>7711997
I don't know if there is a geometric connection but the solution to the basel problem is connected to the circle constant. I can tell you how Euler solved it but I can't give you any insight or intuition as to why the two are related.
>>7711620
>I know nothing, spout authoritative shit about topics I know nothing about, and get angry and post wikipedia links when confronted
do kill yourself
http://www.claymath.org/library/academy/LectureNotes05/Conrad.pdf
