#OK to post homework
#Joseph Koutsoutis, 03-12-2025, Assignment 14pi

#1

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


#After setting Digits := 1000, I found
#evalf(JesusG(10) - 128 / Pi^2) is roughly 1 x 10^(-32)
#evalf(JesusG(100) - 128 / Pi^2) is roughly 4 x 10^(-304)



#2

#This is a Maple implementation of the Chudnovsky algorithm described here: 
#https://en.wikipedia.org/wiki/Chudnovsky_algorithm

binary_split := proc(a,b) local Pab, Qab, Rab, m, Pam, Qam, Ram, Pmb, Qmb, Rmb:
  if b = a+1 then:
    Pab := -(6*a - 5)*(2*a - 1)*(6*a - 1):
    Qab := 10939058860032000 * a^3:
    Rab := Pab * (545140134*a + 13591409):
  else:
    m := floor((a + b) / 2):
    Pam, Qam, Ram := binary_split(a, m):
    Pmb, Qmb, Rmb := binary_split(m, b):
        
    Pab := Pam * Pmb:
    Qab := Qam * Qmb:
    Rab := Qmb * Ram + Pam * Rmb:
  fi:
 Pab, Qab, Rab:
end:


Chudnovsky := proc(n) local P1n, Q1n, R1n:
    P1n, Q1n, R1n := binary_split(1, n):
    426880 * sqrt(10005) * Q1n / (13591409*Q1n + R1n):
end: