#1
#I used the formula given in the "Samples of my Work" Section

JesusG:=proc(N) local T:
T:=sum(((-1)^n)*binomial(2*n, n)^5*(820*n^2+180*n+13)/(2^(20*n)), n=0..N):   
evalf(T);
end: 


#want roughly 12.969

#Even for N as small as 4, this works really well! I got 12.96911151, and the true value (truncated appropriately) is 12.96911150. 

#2:

#used Chudnovsky algorithm wikipedia article

inside:=proc(j)
return(evalf((-(6*j-1)*(2*j-1)*(6*j-5))/(10939058860032000*j^3))):
end:

ChudnovskyBros:=proc(N) local T, j, Acoeff, Baddend, Cprod, D, k, prodfactor:
Acoeff:=1/(426880*evalf(sqrt(10005))):
Baddend:=13591409:
prodfactor:=1:
#splitting up as advised_______
for k from 1 to N do
 prodfactor:=prodfactor*evalf(inside(k)):
od:

Cprod:=545140134*N+13591409:
D:=Acoeff*(Baddend+prodfactor*Cprod):
T:=1/D:
return(evalf(T)):

end: