Taylor & MacClaurin Series
Certain functions can be expressed as power series:

f (x)   an x

 an x
n 0
n
n0
n
 a0  a1x  a2 x  a3 x  a4 x  ...
2
How can we find the values of a0,
a0  f (0)
a1  ?
3
4
a1, a2, a3…?
Plug in 0 for x and watch all the other terms go away
a1  f (0)
Plug in 0 for x again
2
3

f ( x)  a1  2a2 x  3a3 x  4a4 x  ...
a0  f (0)
a1  ?
Plug in 0 for x
a1  f (0)
f ( x)  a1  2a2 x  3a3 x  4a4 x  ...
2
3
f (0)
a2 
a2  ?
2
2


f ( x)  2a2  6a3 x  12a4 x  ...
f (0)
Can you find the pattern for finding an?
a3  ?
a3 
6
f ( x)  6a3  24a4 x  ...
So the power series that converges to f
(x) can be written as…
n
f (0)
an 
n!


f (x)   an x  
n
n 0
n 0
n
f (0) n
x
n!
f (0) 2 f (0) 3 f IV (0) 4
f ( x)  f (0)  f (0) x 
x 
x 
x  ...
2
6
24
This is called the Taylor Series for f (x) centered at x = 0 because we used 0 to find
all the terms.
A series centered at x = 0 is also called a MacClaurin Series.
In class, we will discuss how to generate a series centered at a point other than x = 0
Find the Taylor Series centered at x = 0 for the function
f ( x)  e x
We would start by finding the first few derivatives and
then looking for a pattern.
But in this case the derivatives are easy…


n 0
n
f (0) n
x
n!
f (0)  f (0)  f (0)  f (0)...  1
1 2 1 3 1 4
e  1  x  x  x  x ...
2
3!
4!
x
f (0) 2 f (0) 3 f IV (0) 4
f ( x)  f (0)  f (0) x 
x 
x 
x  ...
2
6
24
Find the Taylor Series centered at x = 0 for the function
f ( x)  e x
We would start by finding the first few derivatives and
then looking for a pattern.
But in this case the derivatives are easy…
f (0)  f (0)  f (0)  f (0)...  1
1 2 1 3 1 4
e  1  x  x  x  x ...
2
3!
4!
x

…and the answer is…
n
x
e 
n  0 n!
x
We can approximate the graph of ex by generating terms of the series…
2
x
e  1 x 
2
x
f (x) = ex








We can approximate the graph of ex by generating terms of the series…
2
x
e  1 x 
2
x
f (x) = ex








We can approximate the graph of ex by generating terms of the series…
2
3
x
x
e  1 x  
2 6
x
f (x) = ex








We can approximate the graph of ex by generating terms of the series…
2
3
4
x
x
x
e  1 x   
2 6 24
x
f (x) = ex
What happens every time we add a term?

The approximation gets better

Where does the overlap appear to be centered?
At

x=0





In class you will need to be prepared to generate other series
centered at any value of x
A number of MacClaurin Series for some familiar functions can be
found on page 477