"""
Exercise 5.39 from "A Primer on..."
Fairly advanced plotting exercise, where
the task is to visualize the evolution 
of a Taylor series approximation as 
we add more terms. The details of creating
the actual animation are not important to 
remember (and there will not be animation
questions on the exam), but the overall exercise
is a useful repetition of many topics, including
function programming, array computing, plotting etc.
"""

import numpy as np
import matplotlib.pyplot as plt
from math import factorial


#a) create the function that does the animation
def animate_series(fk, M, N, xmin, xmax, ymin, ymax, n, exact): 
    x = np.linspace(xmin, xmax, n)
    s = np.zeros_like(x)

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

    plt.plot(x, exact(x))

    lines = plt.plot(x, s)

    for k in range(M, N+1):
        s = s + fk(x, k)
        lines[0].set_data(x, s)
        plt.draw()
        plt.pause(0.3)


#b) call the function to animate the Taylor series for the sine function
def fk_sin(x, k):
    return (-1)**k * x**(2 * k +1) / factorial(2 * k + 1)

animate_series(fk_sin, 0, 40, 0, 13 * np.pi, -2, 2, 201, np.sin)
    
#clear the figure to be ready for next animation:
plt.cla()

#b) call the function again, now to animate the inverse exponential:
def f_exp_inv(x, k):
    return (-x)**k / factorial(k)

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

animate_series(f_exp_inv, 0, 30, 0, 15, -0.5, 1.4, 201, exp_inv)