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Math 104 – Rimmer
12.8 Power Series
A power series is a series of the form
n
n
n
c x c c x c x c x
where:
a ) x is a variable
) The 's are constants called the coefficients of the series.
n
b c
For each fixed , the series above is a series of constants
that we can test for convergence or divergence.
x
A power series may converge for some values of
and diverge for other values of.
x
x
Math 104 – Rimmer
12.8 Power Series
2
0 1 2
n
n
f x = c + c x + c x + + c x +
The sum of the series is a function
whose domain is the set of all x for which the series converges.
f ( ) is reminiscent of a polynomial but it x has infinitely many terms
2
0
n n
n
f x x x x x
∞
=
If all 's 1, we have
n
c =
This is the geometric series with r = x.
The power series will converge for x <1 and diverge for all other x.
1,
1
a
a r x s
r
= = ⇒ =
−
1
1 x
=
−
2
0
1
1 ... ...
1
n n
n
x x x x
x
∞
=
= + + + + + =
−
Math 104 – Rimmer
12.8 Power Series
( ) ( ) ( ) ...
n
n
n
c x a c c x a c x a
− = + − + − +
∑
In general, a series of the form
is called a power series centered at a or a power series about a
We use the Ratio Test (or the Root Test) to find for what values of x the series converges.
( )
This is called the
I.O.C..
interval
of convergence
1
lim 1 for convergence
n
n
n
a
L
a
→∞
solve for x − a to get x − a < R
⇒ − R < x − a < R
⇒ a − R < x < a + R
( )
is called the
R.O.C..
R radius
of convergence
Plug in the endpoints to check for convergence
or divergence at the endpoints.
use square brackets [ or ]
use parentheses ( or )
Math 104 – Rimmer
12.8 Power Series
( )
2
1
1
2
n
n
n
n
n x
∞
=
−
∑
( )
( )
( )
1 2
1
1
2 1
lim lim
n
n n
n
n n n
n n
n
n a x
a n x
→∞ →∞
( ) ( )
( )
( )
2
2
lim
n
n n
n n n
n
n x x
n x
→∞
− x
For convergence, this limit
needs to be less than 1
− x
Now we need to solve
this inequality for x.
⇒ x < ⇒ x < 2
Find the radius of convergence and the interval of convergence.
This is the radius
of convergence.
so, − 2 < x < 2
Plug in 2 and 2 to see if there
is conv. or div. at the endpoints.
x = x = −
x = 2
2
1
1 2
2
n n
n
n
n
∞
=
−
2
1
1
n
n
n
∞
=
= −
2
Diverges by the Test for Divergence
since lim 1 does not exist.
n
n
n
→∞
−
x = − 2
2
1
1 2
2
n n
n
n
n
∞
=
− −
2
n 1
n
∞
=
=
2
Diverges by the Test for Divergence
since lim.
n
n
→∞
= ∞
2
1
1 2
2
n
n
n
n
∞
=
− ⋅ −
=
2
n
( )
Radius of convergence: 2
Interval of convergence: 2, 2
R =
Math 104 – Rimmer
12.8 Power Series
Sometimes the Root Test can be used just as the Ratio Test.
( )
When can be written as , then the Root Test should be used.
n
n n
a b
( )
1
3 5
n
n
n
n
x
n
∞
=
−
∑
( )
1
3 5
n
n
x
n
∞
=
−
=
∑
( )
( ) 3 5
3 5
lim lim lim
n
x
n
n
n n
n n n
x
a
n
−
→∞ →∞ →∞
−
= =
= 0
( )
R.O.C.
I.O.C. ,
< 1
We get convergence
no matter what x is
( )
1
lim 0...... ,
n
n
n
a
R O C I O C
a
→∞
= ⇒ = ∞ ⇒ −∞ ∞
No value of will
make this limit 1
to give divergence
x
( )
or lim 0
n
n
n
a
→∞
the power series only converges for all x
Math 104 – Rimmer
12.8 Power Series
the power series only converges at the point x = a
( )
1
! 7
2
n
n
n
n x
∞
=
−
∑
( ) ( )
1
1
1
lim lim
n
n
n n n
n
a n x
a
→∞ →∞
n!
n
( )
n
x −
( ) ( ) ( )
( )
lim
n
n
n n
n
n n x x
n x
→∞
( )( )
1
2
lim 1 7
n
n x
→∞
= ∞ > 1
We get divergence
for all values of
except at
x
x = a
No value of will
make this limit 1
to give convergence
x
<
{ }
1
lim... 0...
n
n
n
a
R O C I O C a
a
→∞
= ∞ ⇒ = ⇒ ( )
or lim
n
n
n
a
→∞
{ }
R.O.C. 0
I.O.C. 7
at x = a , each term of the series is 0
Math 104 – Rimmer
12.8 Power Series
( ) ( )
( )
2
2
1
1!
2!
n
n
n
n x
n
∞
=
−
∑
( )
( )
( )
( )
2 1
2 1
1
lim lim
n
n
n
n n
n
n
a x
a n
→∞ →∞
( )
n
( )
2
n!
2 n
x
( )
2 n!
( ) ( )
( )
( ) ( )
( )
( )
( )( )( )
2 2
2 2
2 2
lim
n
n
n n
n
n n n x x
n n n x n
→∞
( )
2
2
2
lim 1
n
n n
x
n n
→∞
( )
2
− x
For convergence, this limit
needs to be less than 1
2
⇒ x <
( )
2
− x
2
⇒ x < 4 ⇒ x < 2
This is the radius
of convergence.
Find the radius of convergence.
Radius of convergence: R = 2
( )
2
n + 1 n !
