import numpy as np
import matplotlib.pyplot as plt

#a)
def seq_a(N):
    index_set = range(0,N+1,2)
    a = np.zeros(len(index_set))

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

    return a

#print(seq_a(100))
#print(7/3)

#b)

def seq_b(N):
    index_set = range(N+1)
    d = np.zeros(len(index_set))

    for n in index_set:
        h = 2**(-n)
        d[n] = np.sin(h)/h

    return d

#print(seq_b(50))

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

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


#This call should work as expected
d1 = D(np.sin,0,80)
plt.plot(d1,'o',fillstyle='none')
plt.show()


#d)
"""
The plot resulting from this call
does not look right:

d1 = D(np.sin,0,80)
plt.plot(d1,'o',fillstyle='none')
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.
"""