import numpy as np

def u(n) :
    return (-1)**n/(n+1)
    
def S(n) :
    s = 0
    for k in range(n+1) :
        s += u(k)
    return s

n = 0
while 1/(n+2) >= 1e-6 :
    n += 1
print("n=",n,"et S=",S(n))
    
def v(n) :
    if n%3 == 0 :
        return u(4*(n//3))
    elif n%3 == 1 :
        return u(4*(n//3)+2)
    else :
        return u(2*(n//3)+1)

def T(n) :
    t = 0
    for k in range(n+1) :
        t += v(k)
    return t
    
for n in [10,20,100,500,1000] :
    print(S(n)/T(n))