MATH 214 Lecture Notes
Section 11.8-11.10
1
Power series
Definition: A power series about a (centered at a ) is a series of the form
∞
X
cn (x − a)n = c0 + c1 (x − a) + · + cn (x − a)n + · · ·
n=0
in which the coefficients c0 , c1 , · · · , cn , · · · and the center a are constants, x is a variable.
Example: For what values of x do the following power series converges ?
(a)
∞
X
xn = 1 + x2 + x3 + · · ·
n=0
(b)
(c)
∞
X
xn
x2 x3
=1+x+
+
+ ···
2!
3!
n=0 n!
∞
X
n!xn = 1 + x + 2!x2 + 3!x3 + · · ·
n=0
Solution:
1
2
Radius and interval of convergence
The convergence of the power series
∞
X
cn (x−a)n is described by one of the three possibilities:
n=0
(1) There is a number R > 0 such that the series converges if |x − a| < R and diverges if
|x − a| > R
(2) The series converges absolutely for every x.
(3) The series converges only when x = a.
The number R in case (1) is called radius of convergence of the power series. By convention, the radius of convergence is R = ∞ in case (2) and R = 0 in case (3).
The interval of convergence of a power series is the interval that consists of all values of
for which the series converges.
Example: Find the radius of convergence and the interval of convergence of the power series
∞
X
(−3)n (x − 1)n
√
.
n+1
n=0
Solution:
2
3
Representations of functions as power series
We wish to represent certain types of functions as sums of power series.
We start with the power series
∞
X
xn = 1 + x + x2 + x3 + · · ·
n=0
1
This is a geometric series with ratio x. It converges to
if |x| < 1 and diverges if |x| ≥ 1.
1−x
1
Thus, the function f (x) =
has a power series representation as follows:
1−x
∞
X
1
=
xn = 1 + x + x2 + x3 + · · · ,
1 − x n=0
|x| < 1
Example (substitution): Express the following function as the sum of a power series and
find the interval of convergence.
(a)
1
1+x
(b)
1
1 + x2
3
4
Term-by-term differentiation and integration
Theorem: If the power series
∞
X
(x−a)n has radius of convergence R > 0, then the function
n=0
f defined by
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + · · · =
∞
X
cn (x − a)n
n=0
is differentiable ( and therefore continuous) on the interval (a − R, a + R) and
(1) f 0 (x) = c1 + 2c2 (x − a) + 3c3 (x − a)2 + · · · =
∞
X
ncn (x − a)n−1
n=0
Z
(2)
(x − a)2
(x − a)3
f (x)dx = C + c0 (x − a) + c1
+ c2
+ ···
2
3
∞
n+1
X (x − a)
cn
.
=C+
n+1
n=0
The radii of convergence of the power series in (1) and (2) are both R.
Example: Find a power series representation for the following functions. What are the
intervals of convergence?
(a)
1
(1 + x)2
(b) ln(1 + x)
(c) arctan x
4
5
Taylor series
Suppose that f is a function that can be represented by a power series
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + · · · =
∞
X
cn (x − a)n ,
n=0
which converges in the interval (a − R, a + R) with R > 0. Conducting term-by-term
differentiation n times, we obtain
cn =
f (n) (a)
n!
Consequently, for x ∈ (a − R, a + R) we have
f 00 (a)
f (x) = f (a) + f (a)(x − a) +
(x − a)2 + · · ·
2!
∞
X
f (n) (a)
n
=
(x − a) ,
n!
n=0
0
This series is called the Taylor series of f at a.
The Taylor series about 0 is called the Maclaurin series.
Example: Find the Maclaurin series of the following functions and their radii of convergence.
(a) f (x) = ex
(b) sin x
(c) cos x
5
Example: Find the Maclaurin series of the following functions and their radii of convergence.
2
(a) f (x) = ex (by Substitution)
(b) g(x) =
1 − cos x
(by Arithmatic Operations)
x2
6
Example: Evaluate the indefinite integral
Z
x5 cos(x2 )dx
as an infinite series.
7