4.3. POWER SERIES 145
4.3 POWER SERIES
In this section we study series again. e good news is that we do not have any additional
convergence tests. e bad news is that these series will have variables in them.
Definition 4.12 A power series is a series of the form:
1
X
nD0
a
n
x
n
In a way, a power series is actually an infinite number of different ordinary series, one for each
value of x you could substitute into it. e goal of this section will be: given a power series, find
values of x which cause it to converge.
Knowledge Box 4.21
e radius of convergence of a power series
e power series
1
X
nD0
a
n
x
n
converges in one of three ways:
1. Only at x D 0.
2. For all jxj < r and possibly at x D ˙ r.
3. For all x.
e number r is the radius of convergence of the power series. In the first case
above, we say the radius of convergence is zero; in the third, we say the radius of
convergence is infinite.
Definition 4.13 e interval of convergence of a power series is the set of all x where it converges.
Knowledge Box 4.21 implies that the interval of convergence of a power series is one of Π0; 0 ,
.r; r/, Œr; r/, .r; r, Œr; r, or .1; 1/. e results with an r in them occur in the case
146 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
where 0 < r < 1. Once we have the radius of convergence, in the case where r is positive and
finite, we determine the interval of convergence by checking the behavior of the series when we
set x D ˙ r.
Example 4.54 Find the radius and interval of convergence of:
1
X
nD0
x
n
Solution:
We start by trying to determine the radius of convergence.
Use the ratio test:
lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
x
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
jxj D jxj
is is true because x does not depend on n.
e series thus converges when jxj < 1, meaning we have convergence for sure when
1 < x < 1.
is also means the radius of convergence is r D 1.
We now need to check x D ˙1 to determine the interval of convergence.
ese values both yield non-converging geometric series:
1
X
nD0
.1/
n
and
1
X
nD0
1
So the potential endpoints of the interval of convergence are not part of the interval of
convergence. is means that the interval of convergence is .1; 1/.
˙
4.3. POWER SERIES 147
Example 4.55 Find the radius of convergence of:
1
X
nD0
x
n
2
n
Solution:
Again, use the ratio test.
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
=2
nC1
x
n
=2
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
2
nC1
2
n
x
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
x
2
ˇ
ˇ
ˇ
D
jxj
2
So the series converges when:
1 <
x
2
< 1
2 < x < 2
e radius of convergence is r D 2.
˙
Example 4.56 Find the radius of convergence of the series:
1
X
nD0
x
n
nŠ
148 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
Solution:
Like before:
lim
n!1
ˇ
ˇ
ˇ
ˇ
a
nC1
a
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
=.n C1/Š
x
n
=nŠ
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
n C1
ˇ
ˇ
ˇ
ˇ
D jxj lim
n!1
ˇ
ˇ
ˇ
ˇ
1
n C1
ˇ
ˇ
ˇ
ˇ
D jxj 0
D 0
Since 0 < 1 for all values of x, this power series converges everywhere and the radius of
convergence is infinite.
˙
Example 4.57 Find the radius of convergence of the series:
1
X
nD0
x
n
n
n
Solution:
is time use the root test.
lim
n!1
n
s
ˇ
ˇ
ˇ
ˇ
x
n
n
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
x
n
ˇ
ˇ
ˇ
D jxj lim
n!1
1
n
D 0
Which tells us that, as 0 < 1 for all x, that this sequence converges everywhere.
˙
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