c
Math 152, Benjamin
Aurispa
10.7 Taylor and Maclaurin Series
What if a function cannot be represented using the fundamental geometric series as in the previous section?
Another method is to consruct a Taylor series for the function.
The Taylor Series for a function f (x) about x = a is defined to be
f (x) =
∞
X
f (n) (a)
n=0
n!
(x − a)n = f (a) + f ′ (a)(x − a) +
f ′′ (a)
f ′′′ (a)
(x − a)2 +
(x − a)3 + . . .
2!
3!
where f (n) (a) is the nth derivative of f at x = a.
In particular, if the Taylor series is centered at a = 0, it is referred to as a Maclaurin series and has the
form:
∞
X
f (n) (0) n
f ′′ (0) 2 f ′′′ (0) 3
f (x) =
x = f (0) + f ′ (0)x +
x +
x + ...
n!
2!
3!
n=0
As usual, the Ratio Test can be used to find the radius of convergence for a Taylor or Maclaurin series.
Example: Find the Maclaurin series for f (x) = ex and determine its radius of convergence.
Example: Find the Taylor series for f (x) = e2x centered at a = −4.
1
c
Math 152, Benjamin
Aurispa
Example: Suppose for a function f (x) it is known that f (n) (2) =
f (x) about a = 2.
2n (n + 3)n!
. Find the Taylor series for
5n
Example: Find the Taylor series centered at x = 3 for the function f (x) =
convergence?
2
1
. What is its radius of
3x + 1
c
Math 152, Benjamin
Aurispa
Example: Find the Taylor series for f (x) = ln x centered at x = 5.
Example: Suppose the Taylor series for a function f (x) about x = 7 is given by f (x) =
∞
X
3n (n + 4)
n=0
Find f (10) (7).
3
(n + 1)!
(x−7)n .
c
Math 152, Benjamin
Aurispa
Example: Find the Maclaurin series for f (x) = cos x and determine its radius of convergence.
Example: Find the Maclaurin series for f (x) = sin x and determine its radius of convergence.
4
c
Math 152, Benjamin
Aurispa
YOU MUST MEMORIZE AND KNOW THESE FIVE MACLAURIN SERIES!!!
f (x)
Maclaurin Series
R
I
1
1−x
∞
X
R=1
(−1, 1)
∞
X
xn
R=∞
(−∞, ∞)
∞
X
(−1)n x2n+1
R=∞
(−∞, ∞)
∞
X
(−1)n x2n
R=∞
(−∞, ∞)
R=1
[−1, 1]
xn
n=0
ex
n=0
sin x
n=0
cos x
n=0
arctan x
n!
(2n + 1)!
(2n)!
∞
X
(−1)n x2n+1
n=0
(2n + 1)
Find the Maclaurin series and radius of convergence for f (x) = x sin
Find
Z
cos(x6 ) dx as an infinite series.
5
!
5x3
.
2
c
Math 152, Benjamin
Aurispa
arctan(2x) − 2x
using series.
x→0
x3
Evaluate lim
Evaluate lim
1 − cos 3x
using series.
−1−x
x→0 ex
6
c
Math 152, Benjamin
Aurispa
Evaluate
Z
1
2
e−x dx as an infinite series.
0
If we use the third partial sum to approximate this integral, what is the error in this approximation?
Approximate
Z
1
2
e−x dx correct to within 0.001.
0
7
c
Math 152, Benjamin
Aurispa
Find the sums of the following series.
•
(−1)n π 2n+1
62n+1 (2n + 1)!
n=0
•
∞
X
3(−1)n x5n
∞
X
n!
n=0
•
∞
X
(−4)n π 2n
n=0
• 3+
9n (2n)!
9
2
+
27
6
+
81
24
+
243
120
+ ...
8
c
Math 152, Benjamin
Aurispa
10.9 Taylor Polynomials and Taylor’s Inequality
An nth degree Taylor polynomial, denoted by Tn (x), is just a partial sum of a Taylor series. The Taylor
polynomial of degree n centered at x = a is given by
Tn (x) = f (a) + f ′ (a)(x − a) +
f ′′ (a)
2! (x
− a)2 + · · · +
f (n) (a)
n! (x
− a)n
Taylor polynomials are used to approximate the function f (x) near x = a.
Important Note: Tn (x) is a polynomial of DEGREE n. The number n here does not necessarily refer to the
number of nonzero terms.
Find the 2nd and 3rd degree Taylor polynomials for f (x) = cos 3x centered about a = π6 .
Find T2 (x) for f (x) = x5/4 about a = 16.
9
c
Math 152, Benjamin
Aurispa
Find the 3rd degree Taylor polynomial for xex centered at a = 5.
3
Find T8 (x) for f (x) = x2 e−2x about a = 0.
10