power series
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Power Series
Radii and Intervals of Convergence
First some examples
Consider the following example series:
0
2
!
k
k k
•What does our intuition tell us about the convergence or divergence of this series? •What test should we use to confirm our intuition?
Power Series
Now we consider a whole family of similar series:
0
2
!
k
k k
0
1
!
k
k k
0
4
!
k
k k
0
12!
k
k k
0
214
!
k
k k
•What about the convergence or divergence of these series? •What test should we use to confirm our intuition?
We should use the ratio test; furthermore, we can use the similarity between the series to test them all at once.
0 !
k
k
x
k
How does it go? We start by setting up the appropriate limit.
1!
lim1 !
k
kk
x k
kx
1
1 !lim
!
k
kk
x
k
x
k
lim
1k
x
k
0
Since the limit is 0 which is less than 1, the ratio test tells us that the series
converges absolutely for all values of x.
0 !
k
k
x
k
Why the absolute values?Why on the x’s and not elsewhere?
The series is an example of a power series.0 !
k
k
x
k
What are Power Series?It’s convenient to think of a power series as an infinite polynomial:
Polynomials:
Power Series:
2 3 4
0
3 3 3 3 1 31
3! 5! 7! 9! 2 1 !
k k
k
x x x x x
k
2 311 ( 1) 3( 1) ( 1)4x x x
2 52 3 12x x x
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
In general. . .
Definition: A power series is a (family of) series of the form
00
( ) .nn
n
a x x
In this case, we say that the power series is based at x0 or that it is centered at x0.
What can we say about convergence of power series?
A great deal, actually.
Checking for Convergence
I should use the ratio test. It is the test of choice when testing for convergence of power series!
Checking for Convergence
Checking on the convergence of
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
We start by setting up the appropriate limit.
x1
( 2)lim
( 1)
k
kk
k x
k x
( 2)lim
( 1)k
k x
k
The ratio test says that the series converges provided that this limit is less than 1. That is, when |x|<1.
What about the convergence of
We start by setting up the ratio test limit.
13
2( 1) 1 !lim
3
2 1 !
k
kk
x
k
x
k
13 (2 1)!
lim(2 3)!3
k
kk
x k
kx
2 3 4
0
3 3 3 3 1 31 ?
3! 5! 7! 9! 2 1 !
k k
k
x x x x x
k
3
lim2 2 2 3k
x
k k
3 1 2 3 4 2 1 2 (2 1)lim
1 2 3 4 2 1 2 (2 1) 2 2 2 3k
x k k k
k k k k k
Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely for all x.
0
Now you work out the convergence of
2 3 4
0
3 3 3 3 1 31
3 5 7 9 2 1
k k
k
x x x x x
k
Don’t forget those absolute values!
Now you work out the convergence of
We start by setting up the ratio test limit.
13
2( 1) 1lim
3
2 1
k
kk
x
k
x
k
13 (2 1)
lim(2 3)3
k
kk
x k
kx
2 3 4
0
3 3 3 3 1 31
3 5 7 9 2 1
k k
k
x x x x x
k
What does this tell us?
(2 1)3 lim
(2 3)k
kx
k
3x
•The power series converges absolutely when |x+3|<1. •The power series diverges when |x+3|>1. •The ratio test is inconclusive for x = -4 and x = -2. (Test these separately… what happens?)
Convergence of Power SeriesWhat patterns can we see? What conclusions can we draw?
When we apply the ratio test, the limit will always be either 0 or some positive number times |x-x0|. (Actually, it could be , too. What would this mean?)
•If the limit is 0, the ratio test tells us that the power series converges absolutely for all x.
•If the limit is k|x-x0|, the ratio test tells us that the series converges absolutely when k|x-x0|<1. It diverges when k|x-x0|>1. It fails to tell us anything if k|x-x0|=1.
What does this tell us?
0
1 when | - | the series converges absolutely.x x
k
0
1 when | - | the series diverges.x x
k
0
1 when | - | we don't know.x x
k
Suppose that the limit given by the ratio test is 0| - | .k x x
We need to consider separately the cases when
• k |x-x0| < 1 (the ratio test guarantees convergence),• k |x-x0| > 1 (the ratio test guarantees divergence), and • k |x-x0| = 1 (the ratio test is inconclusive).
This means that . . . Recall that k 0 !
Recapping
0
1 when | - | the series converges absolutely.x x
k
0
1 when | - | the series diverges.x x
k
0
1 when | - | we don't know. x x
k
0x 0
1x
k0
1x k
Must test endpoints separately!
ConclusionsTheorem: If we have a power series ,
• It may converge only at x=x0.
00
( )nn
n
a x x
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
Radius of convergence is 0
Radius of conv. is infinite.
0x 0x R0x R
Radius of conv. is R.
ConclusionsTheorem: If we have a power series ,
• It may converge only at x=x0.
00
( )nn
n
a x x
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
0x 0x R0x R
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