Math 152 Class Notes
November 17, 2015
10.7 Taylor and Maclaurin Series
In this section we study how to nd power series representations for general functions.
Suppose that f is any function and can be represented by a power series
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · ·
Let's determine what the coecients cn must be in terms of f .
Taylor Series
If f has a power series representation centered at a, that is, if
f (x) =
∞
X
cn (x − a)n ,
|x − a| < R
n=0
then its coecients are given by
f (n) (a)
cn =
n!
The series
∞
X
f (n) (a)
n=0
n!
f 00 (a)
f 000 (a)
2
(x − a) = f (a) + f (a)(x − a) +
(x − a) +
(x − a)3 + · · ·
2!
3!
0
n
is called the Taylor
series of the function
f
at
a (or about a
or centered at
a).
For the special case a = 0, the Taylor series becomes
∞
X
f (n) (0)
n=0
n!
f 00 (0) 2 f 000 (0) 3
x = f (0) + f (0)x +
x +
x + ···
2!
3!
n
0
This case arises frequently enough that it is given the special name Maclaurin series.
Not every function has a power series representation. However, the functions we meet frequently, such as polynomial, fractional, exponential, log, trig and
inverse trig functions, do have power series representations. Therefore these functions
are equal to their Taylor series on the interval of convergence.
Remark.
Example 1. Find the Maclaurin series of the function f (x) = ex and its radius of
convergence.
∞ (−1)n 2n x3n
P
Example 2. Find the sum of the series
n!
n=0
Example 3. Find the Taylor series for ex at a = 2. Find the associated radius of
convergence
Example 4. Find the Taylor series for ln x at a = 3. Find the associated radius of
convergence
Example 5. Find the Maclaurin series for sin x. Find the associated radius of convergence
∞
P
(−1)n π 2n+1
Example 6. Find the sum of the series
2n+1 (2n + 1)!
n=0 3
Example 7. Find the Maclaurin series for cos x. Find the associated radius of convergence
Example 8. Find the Maclaurin series for 2x cos(4x2 ). Find the associated radius of
convergence
Example 9. Find the coecient of x2 in the Maclaurin series for
ˆ
Example 10. Evaluate
0
1
e−x dx as a series.
2
ex
.
1−x
1
sin x − x + x3
6
Example 11. Evaluate lim
5
x→0
x
Important Maclaurin series and their interval of convergence
∞
X
1
=
xn = 1 + x + x2 + x3 + · · · ,
1 − x n=0
x
e =
∞
X
xn
n=0
sin x =
x2 x3
=1+x+
+
+ ··· ,
n!
2
3!
∞
X
(−1)n
n=0
∞
X
x2n+1
x3 x5 x7
=x−
+
−
+ ··· ,
(2n + 1)!
3!
5!
7!
x2n
x2 x4 x6
cos x =
(−1)
=1−
+
−
+ ··· ,
(2n)!
2!
4!
6!
n=0
arctan x =
n
∞
X
n=0
(−1)n
x2n+1
x3 x5 x7
=x−
+
−
+ ··· ,
2n + 1
3
5
7
(−1, 1)
(−∞, ∞)
(−∞, ∞)
(−∞, ∞)
[−1, 1]