11
INFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES
In section 11.9, we were able to
find power series representations for
a certain restricted class of functions.
INFINITE SEQUENCES AND SERIES
Here, we investigate more general
problems.
 Which functions have power series
representations?
 How can we find such representations?
INFINITE SEQUENCES AND SERIES
11.10
Taylor and Maclaurin Series
In this section, we will learn:
How to find the Taylor and Maclaurin Series of a function
and to multiply and divide a power series.
TAYLOR & MACLAURIN SERIES
Equation 1
We start by supposing that f is any function
that can be represented by a power series
f ( x)  c0  c1 ( x  a)  c2 ( x  a)
2
 c3 ( x  a)  c4 ( x  a)  ...
3
4
| x  a | R
TAYLOR & MACLAURIN SERIES
Let’s try to determine what the
coefficients cn must be in terms of f.
 To begin, notice that, if we put x = a in
Equation 1, then all terms after the first one
are 0 and we get:
f(a) = c0
TAYLOR & MACLAURIN SERIES
Equation 2
By Theorem 2 in Section 11.9, we can
differentiate the series in Equation 1 term by
term:
f '( x)  c1  2c2 ( x  a)  3c3 ( x  a)
 4c4 ( x  a)  ...
3
2
| x  a | R
TAYLOR & MACLAURIN SERIES
Substitution of x = a in Equation 2
gives:
f’(a) = c1
TAYLOR & MACLAURIN SERIES
Equation 3
Now, we differentiate both sides of Equation 2
and obtain:
f ''( x)  2c2  2  3c3 ( x  a)
 3  4c4 ( x  a)  ... | x  a | R
2
TAYLOR & MACLAURIN SERIES
Again, we put x = a in Equation 3.
 The result is:
f’’(a) = 2c2
TAYLOR & MACLAURIN SERIES
Let’s apply the procedure
one more time.
TAYLOR & MACLAURIN SERIES
Equation 4
Differentiation of the series in Equation 3
gives:
f '''( x)  2  3c3  2  3  4c4 ( x  a)
 3  4  5c5 ( x  a) ... | x  a | R
2
TAYLOR & MACLAURIN SERIES
Then, substitution of x = a in Equation 4
gives:
f’’’(a) = 2 · 3c3 = 3!c3
TAYLOR & MACLAURIN SERIES
By now, you can see the pattern.
 If we continue to differentiate and substitute
x = a, we obtain:
f ( n ) (a )  2  3  4    ncn  n !cn
TAYLOR & MACLAURIN SERIES
Solving the equation for the nth
coefficient cn, we get:
cn 
f
(n)
(a)
n!
TAYLOR & MACLAURIN SERIES
The formula remains valid even for n = 0
if we adopt the conventions that 0! = 1
and f (0) = (f).
 Thus, we have proved the following theorem.
TAYLOR & MACLAURIN SERIES
Theorem 5
If f has a power series representation
(expansion) at a, that is, if

f ( x)   cn ( x  a )
n
| x  a | R
n 0
then its coefficients are given by:
cn 
f
(n)
(a)
n!
TAYLOR & MACLAURIN SERIES
Equation 6
Substituting this formula for cn back into
the series, we see that if f has a power series
expansion at a, then it must be of the following
form.
TAYLOR & MACLAURIN SERIES

Equation 6
f ( n ) (a)
n
f ( x)  
( x  a)
n!
n 0
f '(a)
f ''(a)
 f (a) 
( x  a) 
( x  a) 2
1!
2!
f '''(a)
3

( x  a)  
3!
TAYLOR SERIES
The series in Equation 6 is called
the Taylor series of the function f at a
(or about a or centered at a).
TAYLOR SERIES
Equation 7
For the special case a = 0, the Taylor
series becomes:

f
(n)
(0) n
f ( x)  
x
n!
n 0
f '(0)
f ''(0) 2
 f (0) 
x
x  
1!
2!
MACLAURIN SERIES
Equation 7
This case arises frequently
enough that it is given the special
name Maclaurin series.
TAYLOR & MACLAURIN SERIES
The Taylor series is named after the English
mathematician Brook Taylor (1685–1731).
The Maclaurin series is named for the Scottish
mathematician Colin Maclaurin (1698–1746).
 This is despite the fact that the Maclaurin series is really
just a special case of the Taylor series.
MACLAURIN SERIES
Maclaurin series are named after Colin
Maclaurin because he popularized them
in his calculus textbook Treatise of Fluxions
published in 1742.
TAYLOR & MACLAURIN SERIES
Note
We have shown that if, f can be represented
as a power series about a, then f is equal to
the sum of its Taylor series.
 However, there exist functions that are not
equal to the sum of their Taylor series.
 Give an Example?
TAYLOR & MACLAURIN SERIES
Example 1
Find the Maclaurin series
of the function f(x) = ex and
its radius of convergence.
TAYLOR & MACLAURIN SERIES
Example 1
If f(x) = ex, then f (n)(x) = ex.
So, f (n)(0) = e0 = 1 for all n.
 Hence, the Taylor series for f at 0
(that is, the Maclaurin series) is:


n 0
f ( n ) (0) n  x n
x x 2 x3
x    1     
n!
1! 2! 3!
n 0 n !
TAYLOR & MACLAURIN SERIES
To find the radius of convergence,
we let an = xn/n!
 Then,
an1
x n1 n ! | x |

 n 
 0 1
an
(n  1)! x
n 1
 So, by the Ratio Test, the series converges for all x
and the radius of convergence is R = ∞.