#a)

class Sum:
    def __init__(self,term, M, N):
        self.term = term
        self.M = M
        self.N = N

    def __call__(self,x):
        s = 0
        for k in range(self.M,self.N+1):
            s += self.term(k,x)
        return s


def term(k, x):
    return (-x)**k

#a)
S = Sum(term, M=0, N=3)
x = 0.5
print(S(x))
print(S.term(k=4, x=x))  # (-0.5)**4

#b)
def test_Sum():
    tol = 1e-8

    def term(k, x):
        return (-x)**k

    S = Sum(term, M=0, N=3)
    x = 0.5
    expected = 1 - x + x**2 - x**3
    computed = S(x)
    assert abs(computed-expected) < tol



test_Sum()

#c)
"""
Page 323 in "A primer on..."
Terms in Taylor approximation of sin(x):
(-1)**k*(x**(2*k+1))/factorial(2*k+1)
"""

from math import factorial,pi

def sine_term(k,x):
    return (-1)**k*(x**(2*k+1))/factorial(2*k+1)

for n in range(3,9):
    taylor_n = Sum(sine_term,M=0,N=n)
    print(f"With {n} terms: sin(pi) ~ {taylor_n(pi)}, sin(pi/2) ~ {taylor_n(pi/2)}")

"""
Terminal> python Sum.py
0.625
0.0625
With 3 terms: sin(pi) ~ -0.07522061590362306, sin(pi/2) ~ 0.9998431013994987
With 4 terms: sin(pi) ~ 0.006925270707505135, sin(pi/2) ~ 1.0000035425842861
With 5 terms: sin(pi) ~ -0.00044516023820921277, sin(pi/2) ~ 0.999999943741051
With 6 terms: sin(pi) ~ 2.1142567558399565e-05, sin(pi/2) ~ 1.0000000006627803
With 7 terms: sin(pi) ~ -7.727858894306387e-07, sin(pi/2) ~ 0.9999999999939768
With 8 terms: sin(pi) ~ 2.2419510716912098e-08, sin(pi/2) ~ 1.0000000000000437
"""