Prof of sumfunctions by Abel's theorem

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

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. *