Calculus 141, section 9.8a Power Series & Radius of Convergence
notes by Tim Pilachowski
∞
A series of the form
∑ cn x n = c0 + c1 x + c2 x 2 + K
is called a power series. We’ll be most interested in the
n=0
type for which the coefficients form a sequence, and will ask the question: Does this power series converge?
∞
Example ** (Example G, lecture notes 9.7): Find values of x for which
∑ n x n = x + 2x 2 + K
converges.
n =1
Answer: x < 1
∞
Example A: Determine conditions under which
∑ x n = x + x 2 + K converges.
Answer: x < 1
n=0
Lemma 9.23 gives us conditions for convergence of a power series: For a nonzero real number s,
∞
a. If
∑ cn x n
converges for x = s, then it converges absolutely for x < s .
∑ cn x n
diverges for x = s, then it diverges for x > s .
n=0
∞
b. If
n=0
The “ if ” in part a., is a big one. We have to determine whether or not the power series converges. But notice
∞
that no matter what else might be true, a power series
∑ cn x n
will converge for x = 0. At the other extreme, it
n=0
∞
xn
∑ n! which converges for all values of x (text Example 2).
n=0
Examples ** and A above converge within a specific interval. In other words, we get to Theorem 9.24. For any
power series, exactly one of the following will apply:
is possible to construct a series such as
∞
a.
∑ cn x n
converges only for x = 0, radius of convergence = 0, interval of convergence = [0, 0].
∑ cn x n
converges for all values of x, radius of convergence = ∞ , interval of convergence = (− ∞, ∞ ) .
∑ cn x n
converges for x < R , radius of convergence = R > 0, interval of convergence will be (− R, R ) ,
n=0
∞
b.
n=0
∞
c.
n=0
[− R, R ) , (− R, R ] , or [− R, R ]. See table 9.1 in the text for examples of each.
∞
Example B: Find the interval of convergence of
∑ 2n x n .
n=0
(
Answer: − 1 , 1
2 2
[
∞
)
2n x n
Example C: Find the interval of convergence of ∑
. Answer: − 1 , 1
2 2
n +1
n=0
)
∞
2n x n
Example D: Find the interval of convergence of ∑
. Answer: (–∞, ∞)
n!
n=0
∞
⎛ n + 1⎞
Example E: Find the radius of convergence of ∑ ⎜
⎟
n ⎠
n=0 ⎝
n2
x n . Answer: R = 1
e
Since we can think of power series as “infinite polynomials” with a domain equaling the interval of
convergence, we might ask whether they can be differentiated and integrated. Since derivatives and integrals
can be done term by term, we have the following theorems:
∞
Theorem 9.25: For power series
∑ cn x n
with radius of convergence R, the derivative has the same radius of
n=0
convergence, and
∞
⎞ ∞ d
d ⎛⎜ ∞
n⎟=
n =
c
x
c
x
∑ n ⎟ ∑ dx n
∑ n cn x n −1 for x < R .
dx ⎜ n = 0
n =1
⎝
⎠ n =1
(
)
∞
Theorem 9.28: For power series
∑ cn x n
with radius of convergence R, the integral has the same radius of
n=0
x⎛ ∞
∞ ⎛x
∞
⎞
⎞
c n n +1
n
⎜
⎟
x
for x < R . (We’ll use this theorem,
convergence, and ∫ ∑ c n t dt = ∑ ⎜ ∫ c n t n dt ⎟ = ∑
⎜
⎟
⎟
⎜
n +1
=
=
=
n
0
n
0
n
0
0⎝
⎠
⎝0
⎠
in particular, next time, when we continue section 9.8.).
∞
Example C revisited: Differentiate and integrate
Example F: For any x, e x =
integral.
∞
2n x n
4
∑ n + 1 = 1 + x + 3 x 2 + 2x3 + K
n=0
xn
x 2 x3
1
x
=
+
+
+
+ K Show that this series is its own derivative and
∑ n!
2
!
3
!
n=0