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
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.