Prof of sumfunctions by Abel's theorem

May 2010
6
0
I look for help with a prof :

Let F(x) = \int ((1-t^2)^½) dt
Let F(1) = phi/4

Prove for the series that phi/4 = \sum (n=2 to infinity) [½*3/4 .....(2n-3)( [(2n-2)*(2n+1])

I have the idea that Abel's theorem is relevant for the prof without being able to prove it

Full prof not just a hint.

I am sure that Abel's theorem is useful with changes to allow sum not from n=0 but from n=2.

Help is most wellcome
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
I look for help with a prof :

Let F(x) = \int ((1-t^2)^½) dt
Let F(1) = phi/4

Prove for the series that phi/4 = \sum (n=2 to infinity) [½*3/4 .....(2n-3)( [(2n-2)*(2n+1])

I have the idea that Abel's theorem is relevant for the prof without being able to prove it

Full prof not just a hint.

I am sure that Abel's theorem is useful with changes to allow sum not from n=0 but from n=2.

Help is most wellcome
This is pretty much impossible to read. Could you try to retype it?
 
May 2010
6
0
Carifycation - Sumfunction prove results with Abel's theorem

Let me carify wirrting it more readiable.

Let the Taylor series in x=0 of the function

F(x) = integral of [(1-t^2)]^½
with integral from 0 to x


Observe : F(1) = phi/4

Prof the formula :

Phi/4 = 1 - SUMMATION from n=2 to infinity of (these fractions)

1 3 (2n-2) 1
_ * _ *** *_______
2 4 (2n-2) (2n(2n+1)


I have the idea that Abel's theorem is relevant for the prof without being able to prove it

Full prof not just a hint is needed. Abel's theorem is useful with changes to allow sum not from n=0 but from n=2 to infinity.

FULL Prof is needed to convince me.
 
May 2010
98
16
Im not quite sure, but wouldn't it be an idea to actually see why

\(\displaystyle F(1)= \frac{\pi}{4}\) and then use that to conclude that
the formula holds? But i think your formula is wrong, and that it should be something like \(\displaystyle \frac{\pi}{4} = 1-\frac{1}{6}- \sum_{n=0}^{\infty} \frac {1}{2} \frac {3}{4}...\frac {2n-3}{2n-2} \frac{1}{2n(2n+1)}\)
 
May 2010
6
0
Dialogue - explain Zaph

Zaph

Expalin your point. It's more like a polical option you are coming up with than a math reasoning.

Exlain your point or better give a full proff of your point.

BR
Pragma
 
May 2010
98
16
I'm sorry, i can't help you with a proof, i was just thinking aloud i guess.. Hope you get it sorted, perhaps by people smarter than me :)
 
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