"""
Ex. 5.39 from "A primer on...
Animate how a Taylor series approximation
becomes increasingly accurate as we include more terms.
The approach used here follows Recipe 1 for creating
the animation. 
"""

import numpy as np
import matplotlib.pyplot as plt

#factorial is not in numpy, need import from math:
from math import factorial


#a) Create the function:
def animate_series(fk, M, N, xmin, xmax, ymin, ymax, n, exact):
    x = np.linspace(xmin, xmax, n)
    y_exact = exact(x)
    y = np.zeros_like(x)

    plt.axis([xmin, xmax, ymin, ymax])
    plt.plot(x, y_exact)

    lines = plt.plot(x, y)

    for k in range(M, N+1):
        y = y + fk(x, k)
        lines[0].set_ydata(y)
        plt.draw()
        plt.pause(0.25)


#b) Call the function for the Taylor series approximation to sin(x)
def f_sin(x, k):
    return (-1)**k * x**(2 * k + 1) / factorial(2 * k + 1)
animate_series(f_sin,M=0, N=40, xmin=0, xmax=13*np.pi, ymin=-2, ymax=2, n=100, exact = np.sin) 

#c) Call the function for the Taylor series approximation to exp(-x) 
def f_expinv(x, k):
    return (-x)**k / factorial(k)

def exp_inv(x):
    return np.exp(-x)

animate_series(f_expinv, 0, 30, 0, 15, -0.5, 1.4, 100, exp_inv)