Taylor Series

Came to the Taylor series while I was studying the fundamental element of chaos theory.

So what is Taylor series?

Taylor series is a representation of a function as an infinite sum of terms, calculated from the values of its derivative at a single point ［Wikipedia］.

Let’s look at the definition.

Given a function $f(x)$ and $f(x)$ is differentiable infinitely at a point $a$ of a real or complex domain. Then, we say that, for a value $x$ near $a$ , $f(x)$ can be approximated with Taylor series. That is,

$\displaystyle \sum^{\infty}_{n=0} \frac{f^{(n)}(a) }{n !} (x-a)^n$ .

where $n!$ denotes the factorial of $n$ and $f^{(n)}(a)$ denotes the $n$ th derivatives of $f$ at point $a$ .

OK. Let’s see for an example.

We all know that the derivative of $e^x$ is also $e^x$ . In other words, $e^x$ can be differentiated infinitely.

Every algebra has a value of 1 when power with 0, except $0^0$ which is undefined. So, how about $e^{0.001}$ ?

Of cause we can use calculator to find out the value. But, let’s try with Taylor series method, i.e.

$f(0.001) = \frac{f^{(0)}(0)}{0!}(0.001)^0 + \frac{f^{(1)}(0)}{1!}(0.001)^1 + \cdots + \frac{f^{(n)}(0)}{n!}(0.001)^n$

#
Let’s calculate this with Python.


#python code
import scipy

scipy.e**0

def taylor_euler(a, x, n):


result = 0
for i in range(0, n):
t1  = scipy.e ** (a)
t2 = (x - a) ** i
t3 = scipy.factorial(i)
result += t1 * t2 / t3

return result

print 'Answer of e^0.001 is %0.20f' % scipy.e **0.001


print 'OK. Let''s approximate e^0.001 from e^0 by slowly increase the iterations...'
for i in range(10):
ans  = taylor_euler(0,0.001, i)
print '[%02d] ans = %0.20f' % (i+1, ans)

The output is:


Answer of e^0.001 is 1.00100050016670838460
OK. Lets approximate e^0.001 from e^0 by slowly increase the iterations...
[01] ans = 0.00000000000000000000
[02] ans = 1.00000000000000000000
[03] ans = 1.00099999999999988987
[04] ans = 1.00100049999999995975
[05] ans = 1.00100050016666664021
[06] ans = 1.00100050016670838460
[07] ans = 1.00100050016670838460
[08] ans = 1.00100050016670838460
[09] ans = 1.00100050016670838460
[10] ans = 1.00100050016670838460