Let a function f  x  be given as the sum of a power series in the

convergence interval of the power series f  x    an  x  x0 
n
n 0
Then such a power series is unique and its coefficients are given
by the formula a 
n
f
n
 x0 
n!
If a function f  x  has derivatives of all orders at x0, then we can
formally write the corresponding Taylor series
f  x   f  x0  
f '  x0 
1!
 x  x0  
f ''  x0 
2!
 x  x0 
2

f '''  x0 
3!
 x  x0 
The power series created in this way is then called the Taylor
series of the function f  x  . A Taylor series for
MacLaurin series.
x0  0is called
3

There are functions f (x)
whose formally generated Taylor series do not converge to it.
A condition that guarantees that this will not happen says that
the derivatives of f (x) are all uniformly bounded
in a neighbourhood of x0.
There are functions with a Taylor series that, as a power series,
converges to quite a different function as the following example
shows:
Example
f  x  e

1
x2
for x  0,
f  0  0
For x  0 , we have
  x12 
d e 
1
 2
2
 e x 
f ' x  
dx
x3
2
1
x 3e x
2
and for x = 0:
1
e x 0
1
t
1
x
f '  x   lim
 lim 1  lim 1  lim t 2  lim t 2  0
x 0
x 0
x 0
t 
t 
x
e
2te
2
2
x
x
xe
e

1
2
In a similar way, we could also show that
0  f  0   f '  0   f ''  0  
 f
k
0 
This means that the Taylor series corresponding to f (x)
converges to a constant function that is equal to zero at all
points. But clearly, e

1
x2
 0 for any x  0 .
Taylor series of some functions:
x x 2 x3
e  1   
1! 2! 3!
x
x3 x5 x 7
sin x  x    
3! 5! 7!
x2 x4 x6
cos x  1    
2! 4! 6!
x 2 x3 x 4
ln 1  x   x    
2 3 4