Well I'm abit curious about the Taylor Series-

Suppose f(x)= arctan(x)

Lets find the taylor series of f(x)....

We get that

arctan(x)= SUM n=0 --> infinity [x- x^3/3 + x^5/5 -x^7/7 ..... (-1)^(n+1)*x^(n+1) / (n+1) .......... ]

lets assume that its keeps going and there is no end point k

when we sub in x=1

we get

pi/4= 1-1/3+1/5+1/7 + .................+ 1/(n+2) +.........

now multiply both by 4

we should get pi is approx = 4 -4/2 +4/5 +4/7 ............. + 4/(n+2) +.....

Is this enough ??

is there a proof where it shows pi can be written in rational/irrational numbers etc..

I know my proof seems sort of dodgy- but please correct me

Thanks for your time and effort.