#ex A.1
from numpy import *

#a
def sequence_a(N):
    index_set = range(N+1)
    a = zeros(len(index_set))

    for n in index_set:
        a[n] = (7+1.0/(n+1))/(3-1.0/(n+1)**2)

    return a

#print(sequence_a(100))
#print(7.0/3)


#b
def sequence_b(N):
    index_set = range(N+1)
    D = zeros(len(index_set))
    for n in index_set:
        D[n] = sin(2**(-n))/2**(-n)
    return D

#print(sequence_b(100)[-1])

#c
def D(f,x,N):
    index_set = range(N+1)
    D = zeros(len(index_set))

    for n in index_set:
        h = 2**(-n)
        print(h)
        D[n] = (f(x+h)-f(x))/h

    return D

import matplotlib.pyplot as plt
D1 = D(sin,0,80)
plt.plot(range(80+1),D1,'o')
plt.show()

#d)
#D1 = D(sin,pi,80)
#plt.plot(range(80+1),D1,'o')
#plt.show()
"""
e) The problem is round-off error. If h is smaller than approx 1e-16, pi + h = 0
on a computer. The numerator then becomes exactly zero (to machine precision),
and the fraction is zero even if the denominator is also very small.
"""