Courses
MDE-MET-01; Calculus and Linear Algebra for Graduate Students
Notebooks
Taylor Series

Taylor Series

Taylor Series Expansion

This notebook presents the Taylor series expansion of the sine function, sin⁡(x)\sin\left( x\right) close to zero for an increasing number of terms in the approximation.

∑k=0∞f(k)(c)k!(x−c)k. \begin{equation*} \sum\limits_{k=0}^{\infty} \dfrac{ f^{(k)} \left( c \right) }{k!} \left( x - c \right)^{k}. \end{equation*} sin⁡(x)≈x−x33!+x55!−x77!… \begin{equation*} \sin \left( x \right) \approx x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} \ldots \end{equation*}

Note that in this example, only odd powers of xx contribute to the expansion.

First import the tools needed

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

Define the xx values and set up a figure for plotting

x = np.linspace(-np.pi, np.pi, 200)
plt.figure(figsize = (10,8))

First begin with the first order approximation

n = 0

approximate yy values

y = np.zeros(len(x))

Because only every other term is not equal to zero, we only have to compute n/2n/2 terms

n = n / 2

Taylor series expansion up to nn

for i in np.arange(0, int(n)+1):
  y = y + ((-1)**i * x**(2*i+1)) / factorial(2*i+1)

Plotting

plt.plot(x, y, label = 'First Order')

Third order (same structure as before)

n = 2
y = np.zeros(len(x))
n = n / 2
for i in np.arange(0, int(n)+1):
  y = y + ((-1)**i * (x)**(2*i+1)) / factorial(2*i+1)
plt.plot(x, y, label = 'Third Order')

Seventh order

n = 6
y = np.zeros(len(x))
n = n / 2
for i in np.arange(0,int(n)+1):
  y = y + ((-1)**i * (x)**(2*i+1)) / factorial(2*i+1)
plt.plot(x, y, label = 'Seventh Order')

Eleventh order

n = 10
y = np.zeros(len(x))
n = n / 2
for i in np.arange(0,int(n)+1):
  y = y + ((-1)**i * (x)**(2*i+1)) / factorial(2*i+1)
plt.plot(x, y, label = 'Eleventh Order', marker='.')

Plotting of full solution, i.e. sin⁡(x)\sin \left(x \right)

plt.plot(x, np.sin(x), 'k', label = 'sin(x)')
plt.grid()
plt.title('Taylor Series Approximations of Various Orders for $\sin(x)$')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()

Illustration of convergence of taylor series for sin about zero